AN INTRODUCTION TO GEOMETRIC MEASURE THEORY AND AN …€¦ · AN INTRODUCTION TO GEOMETRIC MEASURE THEORY AND AN APPLICATION TO MINIMAL SURFACES ( DRAFT DOCUMENT) Academic Year 2018/19

AN INTRODUCTION TO GEOMETRIC MEASURE THEORYAND

AN APPLICATION TO MINIMAL SURFACES( DRAFT DOCUMENT)Academic Year 2018/19

Francesco Serra Cassano

ContentsI. Recalls and complements of measure theory.

I.1 Measures and outer measures, approximation of measuresI.2 Convergence and approximation of measurable functions: Severini- Egoroff’s

theorem and Lusin’s theorem.I.3 Absolutely continuous and singular measures. Radon-Nikodym and Lebesgue

decomposition theorems.I.4 Signed vector measures: Lebesgue decomposition theorem and polar decompo-

sition for vector measures.I.5 Spaces Lp(X,µ) and their main properties. Riesz representation theorem.I.6 Operations on measures.I.7 Weak*-convergence of measures. Regularization of Radon measures on Rn.

II. Differentiation of Radon measures.II.1 Covering theorems and Vitali-type covering property for measures on Rn.II.2 Derivatives of Radon measures on Rn. Lebesgue-Besicovitch differentiation

theorem for Radon measures on Rn.II.3 Extensions to metric spaces.

III. An introduction to Hausdorff measures, area and coarea formulas.III.1 Caratheodory’s construction and definition of Hausdorff measures on a metric

space and their elementary properties; Hausdorff dimension.III.2 Recalls of some fundamental results on Lipschitz functions between Euclidean

spaces and relationships with Hausdorff measures.III.3 Hausdorff measures in the Euclidean spaces; H1 and the classical notion of

length in Rn; isodiametric inequality and identity Hn = Ln on Rn; k-dimensionaldensities.

III.4 Area and coarea formulas in Rn and some applications.III.5 Extensions to metric spaces.

IV. Rectifiable sets and blow-ups of Radon measures.IV.1 Rectifiable sets of Rn and their decomposition in Lipschitz images.IV.2 Approximate tangent planes to rectifiable sets.IV.3 Blow-ups of Radon measures on Rn and rectifiability.IV.4 Extensions to metric spaces.

V. An introduction to minimal surfaces and sets of finite perimeter.V.1 Plateau problem: nonparametric minimal surfaces in Rn, area functional and

its minimizers.V.2 Direct methods of the calculus of variations and application to the existence

of minimizers for the Plateau problem.V.3 Sets of finite perimeter, space of bounded variation functions and their main

properties; sets of minimal boundary.

1

2

V.4 Structure of sets of finite perimeter and reduced boundary.V.5 Regularity of minimal boundaries.V.6 Extensions to metric spaces.

SOME BASIC NOTATION

If A, B are sets then the symmetric difference between A and B will be denotedby

A∆B := (A \B) ∪ (B \ A) .

We shall tipically work in a metric space X with a metric d, although we willpresent some notions and results in more general settings. In some chapters howeverwe mainly deal with the Euclidean n- space Rn. Here are the basic notations used inmetric spaces throughout these notes. The closed and open balls with centre x ∈ Xand radius r, 0 < r < ∞, are denoted by

B(x, r) = y ∈ X : d(x, y) 6 r ,

U(x, r) = y ∈ X : d(x, y) < r .In Rn we also set

B(r) = B(0, r), U(r) = U(0, r), S(x, r) = ∂B(x, r) and S(r) = ∂B(0, r);

If B = B(x, r) (respectively B = U(x, r)) and α > 0, we denote αB = B(x, α r)(respectively αB = U(x, α r)). When α = 5 we will call 5B an enlargement of B and

we will denote it by B.

The diameter of a nonempty subset A ⊂ X is

d(A) = diam(A) = sup d(x, y) : x, y ∈ A .We agree d(∅) = 0. If x ∈ X and A and B are non-empty subsets of X, the distancefrom x to A and the distance between A and B are, respectively,

d(x,A) = inf d(x, y) : y ∈ A ,

d(A,B) = inf d(x, y) : x ∈ A, y ∈ B .For ε > 0 the open ε-neighbourhood of A is

Iε(A) = x ∈ X : d(x,A) < ε .If A ⊂ Rn, then

|A| = Ln(A)

where Ln denote the n-dimensional Lebesgue outer measure.

1. Recalls and complements of measure theory([AFP, GZ, Mag, R1, SC]).

Motivation: The main goal is to recall and complement some notions and re-sults of measure theory such as: outer measure, measure, signed measures and vectormeasure with their properties and relationships; measurable functions and their prop-erties; Lp spaces and Riesz representation theorem; convergence of measures.

3

1.1. Measures and outer measures, approximation of measures.

Measures and outer measures.

Let us quickly recall some important notions and results of abstract measure theory(see [GZ]).

Tipically there are two approachs in abstract measure theory: one by using measure,may be more ordinary in the literature, and one by outer measure due to Caratheodory.

Firstly let us introduce the so-called set-theoretic approach where we introduce thenotion of ”measure” and ”measurable set”, only assuming that the environment X isa set.

Definition 1.1. Let X denote a set and P(X) denote the class of all subsets of X.

(i) A set function ϕ : P(X)→ [0,∞] is called an outer measure (o.m.) on X if

(OM1) ϕ(∅) = 0 ,

(OM2) ϕ(A) 6 ϕ(B) if A ⊂ B (monotonicity) ,

(OM3)

ϕ(∪∞i=1Ai) 6∞∑i=1

ϕ(Ai) for any sequence (Ai)i ⊂ X (countable subadditivity) .

(ii) A set E ⊂ X is called ϕ-measurable (with respect to an o.m. ϕ on X), if

ϕ(A) = ϕ(A ∩ E) + ϕ(A \ E) ∀A ⊂ X .

The class of ϕ-measurable sets will be denoted by Mϕ.(iii) A σ-algebra M on X is a (nonempty) class of subsets M ⊂ P(X) satisfying

the two following properties:

(σA1) if E ∈M, then X \ E ∈M ;

(σA2) for each sequence (Ei)i ⊂M, then ∪∞i=1 Ei ∈M .

(iv) A measure µ on X is a set function µ :M→ [0,∞], where M is a σ-algebraon X, satisfying the following two properties:

(M1) µ(∅) = 0 ;

(M2) µ(∪∞i=1Ei) =∞∑i=1

µ(Ei) (countable additivity)

for each disjoint sequence (Ei)i ⊂ M. The structure composed by the triple(X,M, µ), or also the couple (X,M), is called measure space and the setscontained in M are called measurable sets.

(v) Let (X,M, µ) be a measure space. The measure µ is said to be finite, ifµ(X) < ∞; it is said to be σ-finite, if there exists a sequence (Xi)i ⊂M suchthat X = ∪∞i=1Xi and µ(Xi) < ∞ for each i.

(v) Let (X,M, µ) be a measure space. A point x ∈M is said to be an atom if thesingleton x ∈ M and µ(x) > 0. The set of atoms of µ will be denotedby Sµ and µ is said to be atomic if Sµ 6= ∅. If µ is finite or σ-finite the set ofatoms Sµ is at most countable.

4

(vii) A function f : X → R := [−∞,∞] is called measurable with respect to ano.m. ϕ (respectively with respect to a measure µ : M→ [0,∞]) if f−1(U) isϕ-measurable (respectively f−1(U) ∈M) for each open set U in R.

(viii) A simple function s : X → R is one that assumes only a finite number of values.More precisely, s is a simple function if and only if it can be represented as

s(x) =k∑i=1

ai χAi(x) ∀x ∈ X ,

with ai ∈ R, Ai ⊂ X (i = 1, . . . , k), X = ∪ki=1Ai and Ai ∩ Aj = ∅ if i 6= j.

Based on the ideas of H. Lebesgue, it is well known that a theory of an abstractintegration can be carried out on a general measure space (X,M, µ) and we refer to[GZ, Chap. 6] or [R1, Chap. 1] for its complete treatment. In particular, let us recallthat, given a measurable function f : X → [−∞,∞], it is possible to make a senseto the value integral of f with respect to µ, denoted∫

X

f dµ ∈ [−∞,∞] .

When the integral is finite, that is∫Xf dµ ∈ (−∞,∞), f is said to be integrable or

also summable.

Example 1.2. Let (X,M) be a measure space, then we define the following setfunctions on M, which turns out to be measures, as it can easily be proved.

(i) (counting measure) We define the set function # : M→ [0,∞] , #(∅) := 0,#(E) as the cardinality of E if it is finite, #(E) = ∞ otherwise.

(iI) (Dirac measures) With each x ∈ X we associate the set function δx : M →[0,∞] defined by δx(E) := 1 if x ∈ E, δx(E) := 0 otherwise. If (xh)h ⊂ Xand (ch)h ⊂ [0,∞) is a sequence such that the series

∑∞h=1 ch is convergent,

we can define the set function∑∞

h=1 ch δxh : M→ [0,∞](∞∑h=1

ch δxh

)(E) :=

∑h:xh∈E

ch .

Measures of this kind are called purely atomic.

Now, we enrich the environment X, by adding a topology and we require compat-ibility between topology and measure.

Definition 1.3. Let (X, τ) denote a topological space and denote B(X) the σ-algebraof Borel sets of X, i.e. the smallest σ-algebra of X which contains the open and closedsets of X.

(i) An o.m. ϕ on X is called a Borel o.m. if the class of ϕ-measurable setsMϕ ⊃ B(X).

(ii) An o.m. ϕ on X is called a Borel regular o.m. if it is a Borel o.m. and foreach A ⊂ X there exists B ∈ B(X) with B ⊃ A and ϕ(A) = ϕ(B).

(iii) An o.m. ϕ on X is called a Radon o.m. if it is a Borel regular o.m. andϕ(K) < ∞ for each compact set K ⊂ X.

5

(iv) An o.m. ϕ on a metric space (X, d) is called a Caratheodory o.m. (or also ametric o.m.) if

ϕ(A ∪B) = ϕ(A) + ϕ(B) ∀A, B ⊂ X with d(A,B) > 0 ,

where d(A,B) := infd(a, b) : a ∈ A, b ∈ B.(v) A measure µ : M→ [0,∞] on X is called a Borel measure if M = B(X).

(vi ) A measure µ : M → [0,∞] on X is called a Radon measure if it is a Borelmeasure and µ(K) < ∞ for each compact set K ⊂ X.

(vii ) An o.m ϕ on X(respectively a Borel measure µ : B(X) → [0,∞]) is said tolocally finite if for each x ∈ X there exist an open neiborghood Ux of x suchthat ϕ(Ux) < ∞ (respectively µ(Ux) <∞).

Remark 1.4. We stress that the notion of Radon o.m. (respectively Radon measure)in a general topological space (X, τ) may actually differ in the current literature fromone given in Definition 1.3 (iii) (respectively Definition 1.3 (vi)). Indeed it couldbe requested that ϕ (respectively µ) must satisfy to be finite on compact sets andapproximation properties (i) and (ii) of Theorem 1.14. The two notions agree on aseparable, locally compact metric space (X, d) because of Theorem 1.14.

The following basic properties of outer measures are well known (see [GZ]).

Theorem 1.5. (i) Let ϕ be an o.m. on X. Then the class of ϕ-measurable setsMϕ is a σ-algebra on X and ϕ : Mϕ → [0,∞] is a measure.

(ii) If ϕ(N) = 0, then N ∈ Mϕ.(iii) (Caratheodory’s criterion) Let ϕ be a Caratheodory o.m. on a metric space

(X, d). Then ϕ is a Borel o.m.

Remark 1.6. The property of Theorem 1.5 (ii) is characteristic to an outer measurebut is not enjoyed by measures. A measure µ : M→ [0,∞] with the property thatall subsets of sets of µ-measure zero are measurable, is said to be complete and(X,M, µ) is called complete measure space. Not all measures are complete, butthis is not a crucial defect since every measure can easily be completed by enlargingits domain of definition to include all subsets of measure zero, that is by replacingMwith its completion denoted M∗ (see [R1, Theorem 1.36] or [GZ, Theorem 4.45]).

Example 1.7. (i) Let Ln denote the Lebesgue o.m. on Rn. Then Ln is a Radonmeasure on Rn. The classMn ≡MLn is called the class of Lebesgue measur-able sets.

(ii) Let s ∈ [0,∞) and Hs denote the s-dimensional Hausdorff measure on Rn.Then Hs is a Borel regular o.m. on Rn, but it is not a Radon o.m. unlesss > n. We will deeply study these measures in Chapter III.

(iii) Let A be a non Borel set of Rn (why does A exists?); let ϕ : P(Rn)→ [0,+∞]be the set function defined as

ϕ(E) :=

0 if E ⊂ A+∞ if E \ A 6= ∅ .

It is easy to see, by the definition, that ϕ is a Caratheodory outer measureon Rn. However it is not Borel regular because it does not exist a Borel setB ⊃ A such that ϕ(B) = ϕ(A).

6

(iv) Let ϕ : P(Rn)→ [0,+∞] be the set function defined as

ϕ(E) :=

0 if E = ∅1 if E 6= ∅ .

It is easy to see, by the definition, that ϕ is an o.m. on Rn andMϕ = ∅,Rn.In particular, it is not a Borel o. m.

By Theorem 1.5 (i) we see that to every o.m. ϕ on X is associated the measurespace (X,Mϕ, ϕ)

Question: Given a measure space (X,M, µ) is there an associated o.m. µ∗ :P(X)→ [0,∞] such that µ∗ = µ on M ?

There is a simple procedure due to Caratheodory to generate from a measureµ : M→ [0,∞] an outer measure µ∗. Moreover µ∗ is also unique (see [GZ, Theorems4.47 and 4.48]).

Theorem 1.8 (Caratheodory-Hahn extension theorem). Let (X,M, µ) be a measurespace, let

µ∗(E) := inf µ(A) : A ⊃ E, A ∈Mfor each E ⊂ X. Then

(i) µ∗ is an o.m.;(ii) µ∗(A) = µ(A) whenever A ∈M;

(iii) M⊂Mµ∗;(iv) Let N be a σ-algebra with M ⊂ N ⊂ Mµ∗ and suppose that ν is a measure

on N such that ν = µ on M. Then ν = µ∗ on N , provided that µ is σ-finite.

µ∗ is called the o.m. generated by µ.

Let us recall three important results on approximation of measures by open andclosed sets. The first result is also contained in [GZ, Theorem 4.17]).

Theorem 1.9 (Approximation of outer measures by open and closed sets). Let ϕ bea Borel (respectively a Borel regular) o.m. on a metric space (X, d) and let B ⊂ Xbe a Borel set (respectively a ϕ-measurable set).

(i) Suppose that ϕ(B) < ∞, then for each ε > 0 there exists a closed set F ⊂ Bsuch that

ϕ(B \ F ) < ε .

(ii) SupposeB ⊂ ∪∞i=1Vi

where each Vi ⊂ X is an open set with ϕ(Vi) < ∞. Then there is an open setU ⊃ B such that

ϕ(U \B) < ε .

Remark 1.10. Notice that, if ϕ(B) = ∞, then the conclusion of Theorem 1.9 (i)may fail. For instance, consider X = R, ϕ = #, B = (0,+∞). Then, for each closedset F ⊂ (0,+∞), #(B \ F ) = ∞. The conclusion of Theorem 1.9 (ii) may also failby means of the same example if the assumptions are dropped.

The first important consequence of Theorem 1.9 is the following

7

Corollary 1.11. Let ϕ be a Borel (respectively a Borel regular) o.m. on a metricspace (X, d). Suppose there exists a sequence of open sets (Vi)i ⊂ X such that

(?) X = ∪∞i=1Vi with ϕ(Vi) < ∞ ∀ i .

Then for each B ∈ B(X) (respectively B ∈Mϕ)

(i) ϕ(B) = infϕ(U) : U ⊃ B,U open;(ii) ϕ(B) = supϕ(C) : C ⊂ B,C closed.

Proof. See, for instance, [SC, Corollary 1.19].

The second important consequence of Corollary 1.11 and the Caratheodory-Hahnextension theorem (Theorem 1.8) is the following approximation result for Borel mea-sures.

Corollary 1.12 (Approximation of Borel measures by open and closed sets). Con-sider a measure space (X,B(X), µ) where X is a metric space and µ is a Borelmeasure. Suppose that the assumption (?) of Corollary 1.11 holds replacing ϕ withµ. Then for each B ∈ B(X)

(i) µ(B) = infµ(U) : U ⊃ B,U open;(ii) µ(B) = supµ(C) : C ⊂ B,C closed.

Proof. See, for instance, [SC, Corollary 1.20].

When (X, d) is a separable, locally compact metric space and ϕ (respectively µ)is a Radon outer measure (respectively Radon measure) on X, the assumption (?) issatisfied. Thus the conclusions of Corollary 1.11 (respectively Corollary 1.12) hold.Moreover the approximation from below by means of closed sets can be replaced bycompact sets.

Let us recall

Definition 1.13. Let (X, τ) be a topological space.

(i) (X, τ) is said to be separable if it has a countable dense subset.(ii) (X, τ) is said to be locally compact if for each x ∈ X there is an open set

O 3 x such that the closure of O, denoted by O, is compact .

Theorem 1.14 (Approximation of Radon measures on l.c.s. metric spaces). Let(X, d) be a separable, locally compact metric space and ϕ (respectively µ) be a Radonouter measure (respectively Radon measure) on X. Then

(i) for each B ⊂ X, ϕ(B) = infϕ(U) : U ⊃ B,U open(respectively, for each B ∈ B(X), µ(B) = infµ(U) : U ⊃ B,U open);

(ii) for each B ∈Mϕ, ϕ(B) = supϕ(K) : K ⊂ B,K compact(respectively, for each B ∈ B(X), µ(B) = supµ(K) : K ⊂ B,K compact).

Remark 1.15. An immediate consequence of Theorem 1.14 is that, if two Radonmeasures on a locally compact metric space (X, d) agree on the class open set, thenthey have to agree on P(X).

Before the proof of Theorem 1.14 we need the following topological results, whosethe former is well known (see, for instance, [Ro, Proposition 7.6]).

8

Lemma 1.16. Let (X, d) be a separable metric space and let D = xi : i ∈ N ⊂ Xbe dense. Then the family of open sets

U := U(xi, q) : i ∈ N, q ∈ Q ∩ (0,∞)is a basis for the topology induced on X by the distance, where U(x, r) := y ∈ X : d(x, y) < rif x ∈ X and r > 0.

Lemma 1.17. Let (X, d) be a separable, locally compact metric space. Then thereexists an increasing sequence of open sets (Vi)i such that

(1.1) X = ∪∞i=1Vi, Vi is compact for each i .

Proof of Lemma 1.17. Recall that, by definition, (X, d) is locally compact if and only

if ∀x ∈ X ∃ rx > 0 such that U(x, rx) is compact. Let D = xi : i ∈ N ⊂ Xbe dense. From Lemma 1.16, the family U is a basis for the topology and let usenumerate U , that is assume that U = Ui : i ∈ N.Thus there exists a set I(x) ⊂ Nfor which

U(x, rx) = ∪i∈I(x)Ui .

In particular, there exists a choice function α : X → N satisfying:

(1.2) x ∈ Uα(x) and Uα(x) ⊂ U(x, rx) ⊂ U(x, rx) .

Let J := α(X) ⊂ N and

Vi := ∪j∈(J∩1,...,i)Uj if i ∈ N .Then, by (1.2), (1.1) follows.

Proof of Theorem 1.14. Let us first notice that, without loss of generality, we canassume that B ∈ B(X). Indeed, if not, since ϕ is a Borel regular o.m., we can replaceB by a Borel set B ⊃ B and ϕ(B) = ϕ(B). By Lemma 1.17, (?) of Corollary 1.11is satisfied. Thus claim (i) follows at once from Corollary 1.11 (i) (respectively fromCorollary 1.12 (i)). Let us now prove (ii) for a given ϕ Radon o.m. and B ∈ Mϕ.Since each compact set is also closed, from Corollary 1.11 (ii), it follows that

ϕ(B) = supϕ(C) : C ⊂ B,C closed > supϕ(K) : K ⊂ B,K compact .Thus we have only to prove that

(1.3) ϕ(B) = supϕ(C) : C ⊂ B,C closed 6 supϕ(K) : K ⊂ B,K compactLet C ⊂ B be a closed set, let (Vi)i be the sequence of open sets in (1.1) and let

Ki := C ∩ (∪ij=1Vj) .

Then (Ki)i is an increasing sequence of compact sets such that

C := ∪∞i=1Ki .

Observe now that

(1.4) ϕ(Ki) 6 supϕ(K) : K ⊂ B,K compact for each i ,

and, by the continuity of o.m. ϕ on increasing sequences of measurable sets,

(1.5) limi→∞

ϕ(Ki) = ϕ(C) .

By (1.4) and (1.5), (1.3) follows.

9

1.2. Convergence and approximation of measurable functions: Severini-Egoroff’s and Lusin’s theorems.

Theorem 1.18 (Severini-Egoroff). Let (X,M, µ) be a measure space with µ finite.Suppose fh : X → R (h = 1, 2, . . . ) and f : X → R are measurable functions that arefinite µ-a.e.on X. Also, suppose that (fh)h converges pointwise µ-a.e. to f . Then foreach ε > 0 there exists a set A ∈ M such that µ(X \ A) < ε and fh → f uniformlyon A, that is

supx∈A|fh(x)− f(x)| → 0 as h→∞ .

.

Proof. See [GZ, Theorem 5.15].

Remark 1.19. The hypothesis that µ(X) < ∞ is essential in Severini-Egoroff’stheorem. Consider the case of Lebesgue measure L1 on R and define a sequence offunctions by

fh = χ[h,∞) ,

for each positive integer h. Then, limh→∞ fh(x) = 0 for each x ∈ R, but (fh)h doesnot converge uniformly to 0 on any set A whose complement R\A has finite Lebesguemeasure. Indeed, it would follow that R \ A does not contain any half-line [h,∞);that is, for each h, there would exist x ∈ [h,∞)∩A withfh(x) = 1, thus showing that(fh)h does not converge uniformly to 0 on A.

Theorem 1.20 (Approximation by simple functions). Let (X,M) be a measure spaceand let f : X → [0,+∞] be a measurable function. Then there exists a sequence ofmeasurable simple functions sh : X → [0,+∞) (h = 1, 2, . . . ) satisfying the proper-ties:

(i) 0 6 s1 6 s2 6 . . . 6 sh 6 . . . 6 f ;(ii) limh→∞ sh(x) = f(x) ∀x ∈ X.

In particular, if∫Xf dµ < ∞, then∫

X

|f − sh| dµ→ 0 .


Let us now introduce two spaces of continuous functions which play an importantrole in measure theory.

Definition 1.21. Let (X, τ) be a topological space.

(i)

C0c(X) := f : X → R : f is continuous and spt(f) is compact in (X, τ)

where

(1.6) spt(f) := closure x ∈ X : f(x) 6= 0 .

(ii)

C0b(X) := f : X → R : f is continuous and bounded

10

Remark 1.22. Observe that, if a topological space (X, τ) is not locally compact, thespace C0

c(X) could turn out to be meaningless, that is C0c(X) = 0.Indeed

Exercise: Let (X, ‖ · ‖) be an infinite-dimensional normed vector space. ThenC0c(X) = 0.

Lusin’s theorem 1.23 (1912, form on locally compact metric spaces). Let µ be aRadon outer measure on a locally compact, separable metric space X . Let f : X → Rbe a µ-measurable function such that there exists a Borel set A ⊂ X with

µ(A) < ∞, f(x) = 0 ∀x ∈ X \ A and |f(x)| < ∞ µ− a.e. x ∈ X .

Then, for each ε > 0, there exists g ∈ C0c(X) such that

µ (x ∈ X : f(x) 6= g(x)) < ε .

Moreover g can be chosen such that

supx∈X|g(x)| 6 sup

x∈X|f(x)| .

Proof. See [R1, Theorem 2.23].

A consequence of Lusin’s theorem is the following useful approximation of measur-able functions by means of Borel functions.

Corollary 1.24. Let µ be a Radon outer measure on a locally compact, separablemetric space X and let f : X → R be a µ-measurable function. Then there exist aBorel function g : X → R such that f = g µ-a.e. on X.

Proof. See, for instance, [Fe, 2.3.6].

1.3. Absolutely continuous and singular measures. Radon-Nikodym andLebesgue decomposition theorems. Firstly, let us introduce some definitions andpreliminary results.

Definition 1.25. Let (X ,M) be a measure space and let µ , ν : M→ [0,∞] be twomeasures.

(i) The measure ν is said to be absolutely continuous with respect to the measureµ, written ν << µ, if it holds that

µ(E) = 0⇒ ν(E) = 0 .

(ii) The measures ν and µ are said to be mutually singular, written µ ⊥ ν, ifthere exists a measurable set E such that

ν(E) = µ(X \ E) = 0 .

The following result justifies why the word ”continuity” is used in this context.

Theorem 1.26. Let ν be a finite measure and µ a measure on a measure space(X,M). Then the following are equivalent:

(i) ν << µ;(ii) limµ(A)→0 ν(A) = 0, that is, for every ε > 0 ∃ δ = δ(ε) > 0 such that ν(E) < ε

whenever µ(E) < δ.


11

Let us now introduce two fundamental results of measure theory: the Radon-Nikodym and Lebesgue’s decomposition theorems.

Let (X ,M, µ) be a measure space and w : X → [0,∞] be measurable. Thenit is easy to see that µw is a measure, absolutely continuous w.r.t. µ. The veryremarkable fact, content of the Radon-Nikodym theorem, is that essentially eachmeasure ν, absolutely continuous w.r.t. µ is of this form.

Theorem 1.27 (Radon-Nikodym). Let ν and µ be two measures on (X,M). Supposethat

(i) ν and µ are σ-finite, that is, there exists a sequence (Xi)i ⊂ M such thatX = ∪∞i=1Xi and

ν(Xi) < ∞ and µ(Xi) < ∞ for each i .

(ii) ν << µ.Then there exists a measurable function w : X → [0,∞] such that ν = µw

on M , that is,

(RN) ν(E) = µw(E) :=

∫E

w dµ ∀E ∈M .

Moreover the function w in (RN) is µ-a.e. unique.

Definition 1.28. The function w in (RN) is called the Radon-Nikodym derivative of

ν with respect to µ and denoted by w =dν

dµ.

Remark 1.29. Because ν is σ-finite, then w is also σ-integrable with respect to µ,that is,

(σI) 0 6∫Xi

w dµ < ∞ ∀ i ,

where (Xi)i is the sequence in statement (i).

Proof. See See [GZ, Theorem 6.38] and also [SC, Theorem 1.30].

Remark 1.30. We can actually weaken the assumptions of the Radon-Nikodymtheorem. Indeed it is sufficient to require that only µ is σ-finite in order that (RN)holds (see, for instance,[Ro, Theorem 23, Chap. 11]). When µ is not σ-finite, theRadon- Nikodym theorem fails (see Exercise I.8).

For Radon measures on locally compact, separable metric spaces, the Radon-Nikodym theorem has the following simpler and stronger version.

Theorem 1.31 (Radon-Nikodym’s theorem for Radon measures). If X is supposedto be a locally compact, separable metric space and ν and µ are Radon measures on Xwith ν << µ, then (RN) holds and the Radon-Nikodym derivative w := dν

dµis locally

integrable on X, i.e. w ∈ L1loc(X,µ), where

L1loc(X,µ) :=

f : X → R : f is measurable,

∫K

|f | dµ < ∞ for each compact K ⊂ X

.

Proof. See [SC, Theorem 1.35].

12

Historical notes: The first version of Radon-Nikodym’s theorem is due to H.Lebesgue ([Le]) and to G. Vitali ([Vitali]) when X = R in 1904. Radon extended theresult when X = Rn ([Ra]) in 1913. Eventually Nikodym ([Ni]) extended the resultto the abstract setting in 1930.

A consequence of the Radon-Nikodym theorem is the following

Lebesgue decomposition theorem 1.32. Let ν and µ be σ-finite measures on ameasure space (X,M). Then there is a decomposition of ν such that

ν = νac + νs ,

where νac and νs are still measures on (X,M) with νac << µ and νs ⊥ µ. Thedecomposition is unique.

Proof. See [GZ, Theorem 6.39] and also [SC, Theorem 1.36].

Exercise I.10 Let consider µ = L1, ν = δ0 as measures on the σ- algebra M1

of Lebesgue measurable sets in R,where δ0 denotes the Dirac measure at 0, that is,

δ0(E) :=

1 if 0 ∈ E0 if 0 /∈ E . . Prove that the Lebesgue decomposition of ν with respect

to µ, ν = νac + νs, is given by νac ≡ 0 and νs = ν.

1.4. Signed vector measures. Lebesgue decomposition theorem still holds for amore general class of measures. Namely for set functions ν : M → R which stillverify basic properties of countable additivity.

Definition 1.33 (Signed measures). Let (X,M) be a measure space.

(i) An extended real valued set function ν : M → R is a signed measure if itsatisfies the following three properties:

(SM1) ν assumes at most one of the values +∞, −∞;(SM2) ν(∅) = 0;(SM3) For each sequence of disjoint sets (Ei)i ⊂M, it holds that

ν(∪∞i=1Ei) =∞∑i=1

ν(Ei)

where the series on the right either converges absolutely or diverges to −∞or +∞.

(ii) A signed measure ν : M→ R is said to be absolutely continuous with respectto µ :M→ [0,∞] , written ν << µ, if ν(E) = 0 whenever µ(E) = 0.

(iii) Two signed measure ν, µ : M→ R are said to be mutually singular , writtenν ⊥ µ, if there is E ∈M such that ν(E) = µ(X \ E) = 0.

(iv) A signed measure ν : M → R is said to be finite (respectively σ-finite)if |ν(X)| < ∞ (respectively there exists a sequence (Xi)i ⊂ M such thatX = ∪∞i=1Xi and |ν(Xi)| < ∞ for each i).

(v) A real valued signed measure ν : M → R, that is, if ν(E) ∈ R for eachE ∈M, is called a real measure.

13

Example 1.34 (Examples of signed measures). Let us introduce below two remark-able examples of signed measures on a given measure space (X,M).

(i) Let µ : M → [0,∞] be a measure and let f : X → R be a measurablefunction. Suppose at least one of f+ := f ∨ 0 or f− := (−f)∨ 0 is integrable,and let ν : M → R denote the extended real-valued function on M definedby

ν(E) :=

∫E

f dµ ∀E ∈M .

Then is easy to see that ν is a signed measure and ν << µ. If both f+ andf− are integrable, or, equivalently, |f | is integrable, then ν is a real measure.

(ii) Let µ1, µ2 : M → [0,∞] be measures and assume that al least one of themis finite. Let ν : M → R denote the extended real-valued function on Mdefined by

ν(E) := µ1(E)− µ2(E) ∀E ∈M .

Then is easy to see that ν is a signed measure. If both µ1 and µ2 are finite,then ν is a real measure.

Remark 1.35. Observe that a measure is a signed measure. In some contexts wewill emphasize that a measure µ is not a signed measure by saying that it is apositive measure. Notice also that a signed (or also real) measure ν is not an in-creasing set function.

Exercise: A signed measure ν is a real measure if and only if it is finite, that is|ν(X)| <∞.

Theorem 1.36 (Lebesgue decomposition theorem for signed measures). Let (X,M, µ)be a measure space with µ σ-finite, and ν : M → R be a σ-finite signed measure .Then there are two signed measures νac, νs : M→ R such that

(LD) νac << µ, νs ⊥ µ, ν = νac + νs ,

and there exists a measurable function w : X → R such that either w+ or w− isintegrable with respect to µ such that

(RN) νac(E) =

∫E

w dµ ∀E ∈M .

Moreover both decomposition (LD) and representation (RN) are unique.

Proof. See [F, Theorem 3.8].

Remark 1.37. Notice that the sum of signed measures νac + νs is well defined in(LD) since νac and νs are mutually singular.

Remark 1.38. Suppose ν is a real measure (observe that ν is also σ-finite), that isν : M→ R, and ν << µ with µ a given σ-finite positive measure on M. ApplyingTheorem 1.36, we have that

ν(E) = νac(E) =

∫E

w dµ ∀E ∈M ,

and w : X → R is now an integrable function on X with respect to µ, that is∫X|w| dµ < ∞. Indeed, since ν(E) ∈ R for each E ∈ M, it is easy to see that w

must be integrable.

14

We recommend [GZ, Section 6.5] and [F, Chap. 6] for a complete treatment con-cerning signed measures. However we point out that signed measures in Example1.34 are really the only examples: every signed measure can be represented in eitherof these two forms.

An other important tool in GMT will turn out to be the notion of signed vectormeasure, which is an extension of the one of signed measure.

Definition 1.39 (Vector signed measures). Let (X,M) be a measure space.

(i) A vector set function ν = (ν(1), . . . , ν(m)) : M → Rmis a vector signed

measure if its components ν(i) : M → R (i = 1, . . . ,m) are signed measures(according to Definition 1.33).

(ii) A vector signed measure ν : M→ Rmis a vector measure if it is Rm-valued

vector measure, that is ν : M→ Rm.(iii) If ν is a signed vector measure, we define its total variation |ν| : M→ [0,∞]

as follows:

|ν|(E) = sup

∞∑h=1

|ν(Eh)| : (Eh)h ⊂M pairwise disjoint, E = ∪∞h=1Eh

,

where

|v| :=|v|Rm if v ∈ Rm

∞ if v ∈ Rm \ Rm .

(iv) If ν is a real measure, that is ν : M→ R, we define its positive and negativeparts respectively as follows:

ν+ =|ν|+ ν

2and ν− =

|ν| − ν2

Notation: In the following, we will say countable partition of a set E a pairwisedisjoint sequence of sets (Eh)h such that ∪∞h=1Eh = E.

Remark 1.40. Observe that, according to Definition 1.33 (i), a Rm-valued signed

vector measure ν = (ν(1), . . . , ν(m)) satisfies the following two properties:(SVM1) ν(∅) = 0 := (0, . . . , 0) ∈ Rm

;(SVM2) For each sequence of disjoint sets (Eh)h ⊂M, it holds that

ν(∪∞h=1Eh) =∞∑h=1

ν(Eh) :=

(∞∑h=1

ν(1)(Eh), . . . ,∞∑h=1

ν(m)(Eh)

),

where the series on the right-hand side∑∞

h=1 ν(i)(Eh) (i = 1, . . . ,m) either con-

verges absolutely or diverges to −∞ or +∞.Notice also that, if ν is Rm-valued vector measure, then the absolute convergence

in the series in (SVM2) is a requirement on the set function ν: in fact the sum of theseries cannot depend on the order of its terms, as the union does not. Observe alsothat, when m = 1, the notion of signed vector measure (respectively vector measure)agrees with the one of signed measure (respectively real measure). Eventually noticethat a Rm-valued set function ν = (ν1, . . . , νm) :M→ Rm is a vector measure if andonly if νi :M→ R (i = 1, . . . ,m) is a real measure.

15

Remark 1.41. The introduction of the notion of total variation solves the problemof finding a positive measure µ which dominates a given signed vector measure ν onM in the sense that |ν(E)| 6 µ(E) for each E ∈ M, looking for keeping µ as smallas we can. Every solution to this problem (if there is one at all) must satisfy

µ(E) =∞∑h=1

µ(Eh) >∞∑h=1

|ν(Eh)|

for each partition (Eh)h of any set E ∈M, so that µ(E) is at least equal to quantity|ν|(E). This suggest the reason of total variation’s definition like in Definition 1.39(iii).

Let us show that the total variation of a signed vector measure (respectively vectormeasure) is a positive measure (respectively positive finite measure).

Theorem 1.42. (i) Let ν be a signed vector measure on (X,M). Then its totalvariation |ν| is a positive measure.

(ii) If ν is a vector measure, then |ν| is a positive finite measure, that is |ν|(X) <∞.

Proof. (i) We have to prove that

(1.7) |ν|(∅) = 0

and

(1.8) |ν| is countable additive .

It is trivial, by definition, that (1.7) holds. Let us show (1.8).Let us firstly observe that |ν| : M→ [0,∞] is increasing , that is

(1.9) |ν|(E) 6 |ν|(F ) if E ⊂ F .

Indeed let (Eh)h ⊂M be a partition of E, then the family of sets Eh : h∪F \Eis a countable partition of F . Thus

∞∑h=1

|ν(Eh)| 6∞∑h=1

|ν(Eh)|+ |ν(F \ E)| 6 |ν|(F ) .

Then, taking the supremum on the partitions of E in the previous inequality, we get(1.9). Let us now show that

(1.10) |ν| is countably subadditive

and

(1.11) |ν| is additive .

Observe that, from (1.9), (1.10) and (1.11), (1.8) follows. Indeed let (Eh)h ⊂ M bepairwise disjoint. Then by countable subadditivity (1.10),

(1.12) |ν|(∪∞h=1Eh) 6∞∑h=1

|ν|(Eh) .

On the other hand, by (1.9) and (1.11),

16

(1.13) |ν|(∪∞h=1Eh) > |ν|(∪mh=1Eh) =m∑h=1

|ν|(Eh) ∀m ∈ N .

Thus, by (1.12) and (1.14), (1.8) follows. Let us now prove (1.10). Let E, (Eh)h be inM such that E ⊂ ∪∞h=1Eh. Let us define a pairwise disjoint sequence (E ′h)h ⊂M inthe following way E ′0 := E0 and E ′h := Eh \ ∪h−1

i=1 Ei if h > 1. Let (Fi)i be a partitionof E, since (E ′h ∩ Fi)h is a partition of Fi for fixed i, by countable additivity (SVM2)and (1.9), we can infer

∞∑i=1

|ν(Fi)| =∞∑i=1

∣∣∣∣∣∞∑h=1

ν(E ′h ∩ Fi)

∣∣∣∣∣ 6∞∑i=1

∞∑h=1

|ν(E ′h ∩ Fi)|

=∞∑h=1

∞∑i=1

|ν(E ′h ∩ Fi)| 6∞∑h=1

|ν|(E ′h ∩ E) 6∞∑h=1

|ν|(Eh) .

Taking the supremum on the partitions of E in the previous inequality, it follows that

|ν|(E) 6∞∑h=1

|ν|(Eh)

and (1.10) follows. Let us now prove (1.11). Let E, F ∈ M be disjoint. If at leastone between |ν|(E) and|ν|(F ) is ∞, then, by (1.9), it is immediate that

|ν|(E ∪ F ) =∞ = |ν|(E) + |ν|(F ) .

Thus, without loss of generality, we can assume that both |ν|(E) and|ν|(F ) are finite.By definition, for each ε > 0, there exist a partition (Eh)h ⊂ M of E and one(Fh)h ⊂M of F such that

|ν|(E) 6∞∑h=1

|ν|(Eh) + ε, |ν|(F ) 6∞∑h=1

|ν|(Fh) + ε .

Observe now that the family of sets Eh : h ∈ N ∪ Fh : h ∈ N is a countablepartition of E ∪ F . Then, by the previous inequality, it follows that, for each ε > 0,

|ν|(E) + |ν|(F )− 2ε 6∞∑h=1

|ν|(Eh) +∞∑h=1

|ν|(Fh) 6 |ν|(E ∪ F ) .

By countable subadditivity (1.10) and the previous inequality, (1.11) follows.(ii) It is sufficient to assume that ν is a real measure, that is m = 1 and ν : M→ R.The Rm-valued case being an easy consequence of the following estimate

|ν|(E) 6m∑i=1

|νi|(E) ∀E ∈M ,

if ν = (ν1, . . . , νm). Suppose that for some E ∈ M has |ν|(E) = ∞. Let us thenprove there exist two disjoint sets A, B ∈M such that

(1.14) E = A ∪B, |ν(A)| > 1, |ν(B)| > 1, either |ν|(A) =∞ or |ν|(B) =∞.

17

By definition, there is a partition (Eh)h of E such that

(1.15)m∑h=1

|ν(Eh)| > 2(|ν(E)|+ 1) .

Let I := 1 6 h 6 m : ν(Eh) > 0 and J := 1 6 h 6 m : ν(Eh) < 0. Since, bythe additivity

m∑h=1

|ν(Eh)| =∑h∈I

ν(Eh)−∑h∈J

ν(Eh) = ν(∪h∈IEh)− ν(∪h∈JEh) ,

by (1.15), we can infer that either |ν(∪h∈IEh)| = ν(∪h∈IEh) > (|ν(E)| + 1) or|ν(∪h∈JEh)| = −ν(∪h∈JEh) > (|ν(E)| + 1). Let A denote one between sets ∪h∈IEhand ∪h∈JEh such that |ν(A)| > (|ν(E)|+ 1) and let B := E \ A. Then

|ν(B)| = |ν(E)− ν(A)| > |ν(A)| − |ν(E)| > 1 .

By the additivity of |ν|, it is clear that either |ν|(A) =∞ or |ν|(B) =∞. Therefore(1.14) follows. Now if |ν|(X) = ∞, then we can apply (1.14) with E = X and splitX into two sets A1 and B1 with |ν(A1)| > 1 and |ν|(B1) = ∞. Split B1 into twosets A2 and B2 with |ν(A2)| > 1 and |ν|(B2) = ∞. Continuing in this way, we geta countably infinite disjoint family of sets (Ah)h with |ν(Ah)| > 1 for each h. Thecountable additivity of ν implies that

ν(∪∞h=1Ah) =∞∑h=1

ν(Ah) .

But this series cannot converge since ν(Ah) does not tend to 0 as h → ∞. Thiscontradiction shows that |ν|(X) <∞.

Remark 1.43. The above theorem shows that for any real measure ν, its positiveand negative part are positive finite measures, hence the decomposition ν = ν+− ν−holds; it is known as the Jordan decomposition of ν. We point out that a Jordandecomposition still hold for signed measures, by means of a suitable notion of positiveand negative parts for a signed measure (see [GZ, Theorem 6.31] or [F, Theorem 3.4]).

Corollary 1.44. Let ν : M → R be a signed measure. Then ν is σ- finite if andonly if so does its total variation |ν| : M→ [0,∞].

Proof. If |ν| is σ-finite, since

|ν(E)| 6 |ν|(E) ∀E ∈M ,

according to Definition 1.33 (iv), ν is also σ-finite. Suppose that ν is σ- finite, that isthere is a disjoint sequence (Xk)k ⊂M such that |ν(Xk)| <∞ for each k. For givenk, let us define the set function νk : M→ R defined by νk(E) := ν(E ∩Xk). Noticethat νk is a real measure. Thus, from Theorem 1.42 (ii), its total variation |νk| is apositive finite measure, that is |νk|(X) < ∞. Let us now prove that

(1.16) |νk|(X) = |ν|(Xk)

18

from which it will follow that ν is σ-finite and the proof is accomplished. By definition

|νk|(X) = sup

∞∑h=1

|ν(Eh ∩Xk)| : (Eh)h ⊂M partition of X

6 sup

∞∑h=1

|ν(Fh| : (Fh)h ⊂M partition of Xk

= |ν|(Xk) .

(1.17)

Let (Fh)h ⊂M be a partition of Xk and define the partition of X as E1 := X \Xk,Eh := Fh−1 if h > 2.Then it trivial that

(1.18)∞∑h=1

|ν(Fh)| =∞∑h=1

|ν(Eh ∩Xk)| =∞∑h=1

|νk(Eh)| 6 |νk|(X) .

Therefore, by (1.17) and (1.18), (1.16) follows.

Remark 1.45. It is immediate to check that Rm-valued vector measures can be addedand multiplied by real numbers, hence they form a real vector space; moreover, aneasy consequence of Theorem 1.42 is that the total variation is a norm on the space ofmeasures, which turns out to be a Banach space . If X is a locally compact separablemetric space, it will be identified with the dual of a space of continuous functionsand this will give the completeness in another way (see Corollary 1.78 and Theorem1.83).

Example 1.46. According to the notation for positive measures (see (RN)), given ameasure space (X,M, µ) and a vector function w = (w1, . . . , wm) : X → Rm

, witheach wi : X → R (i = 1, . . . ,m) measurable functions such that either wi,+ or wi,− is

integrable. Let us define the vector set function µw : M→ Rmdefined as follows

(1.19) µw(E) =

∫E

w dµ :=

(∫E

w1 dµ, . . . ,

∫E

wm dµ

)E ∈M .

Then it is easy to see that µw is a signed vector measure and its total variation iscomputed in the following proposition.

Proposition 1.47. Let (X,M, µ) be a measure space and let w = (w1, . . . , wm) :X → Rm

, with each wi : X → R (i = 1, . . . ,m) measurable functions such that eitherwi,+ or wi,− is integrable. Consider the vector signed measure µw in (1.19).

Then

(1.20) |µw|(E) =

∫E

|w| dµ ∀E ∈M .

Proof. It is easy to see that, by definition of total variation for a vector measure,

|µw|(E) 6∫E

|w| dµ ∀E ∈M .

Let us prove the reverse inequality. Let E ∈ M. If |µw|(E) = ∞ we are done.Then suppose |µw|(E) < ∞. Note that, from this assumption, we have that each wi(i = 1, . . . ,m) is integrable on E, that is

∫E|wi| dµ <∞. Without loss of generality,

we can assume that each wi is a real-valued function on E and then we can considerw = (w1, . . . , wm) : E → Rm.

19

Let D = zh : h ∈ N be a dense set in the unit sphere Sm−1 := y ∈ Rm : |y| = 1.For any ε ∈ (0, 1) let us define σ : E → N

σ(x) := min h ∈ N : 〈w(x), zh〉 > (1− ε)|w(x)| x ∈ E ,and let Eh := σ−1(h). Then (Eh)h ⊂ M is a pairwise disjoint sequence and E =∪∞h=0Eh. Therefore

(1− ε)∫E

|w| dµ =∞∑h=0

∫Eh

(1− ε)|w| dµ 6

∞∑h=0

∫Eh

|〈w(x), zh〉| dµ 6∞∑h=0

∫Eh

|w(x)| dµ 6 |µw|(E)∀ε ∈ (0, 1) .

Thus, getting ε→ 0 in the previous inequality, the proof is accomplished.

Definition 1.48 (Integrals). Let (X,M) be a measure space.

(i) Let ν :M→ R be a real measure.If u : X → R is a |ν|-measurable function, we say that u is ν-integrable if u

is |ν|-integrable and we set∫X

u dν :=

∫X

u dν+ −∫X

u dν− .

If u = (u1, . . . , uk) : X → Rkis a |ν|-measurable vector function,we say that

u is ν-integrable if each its component ui (i = 1, . . . , k) is |ν|-integrable andwe set ∫

X

u dν :=

(∫X

u1 dν, . . . ,

∫X

uk dν

).

(ii) Let ν = (ν1, . . . , νm) :M→ Rm be a vector measure.If u : X → R is a |ν|-measurable function, we say that u is ν-integrable if u

is |ν|-integrable and we set∫X

u dν :=

(∫X

u dν1, . . . ,

∫X

u dνm

).

(iii) Let E ∈M, the integral of of a function u on E is defined by∫E

u dν :=

∫X

uχE dν ,

provided that the right-hand side makes sense.

Remark 1.49. Notice that an immediate consequence of the above definition is theinequality ∣∣∣∣∫

X

u dν

∣∣∣∣ 6 ∫X

|u| d|ν|

which holds for every extended real or vector valued summable function u and forevery positive, real or vector measure ν.

Definition 1.50 (Absolute continuity and singularity for signed vector measures).Let (X,M) be a measure space.

20

(i) Let µ be a positive measure and ν be a vector signed measure on the measurespace (X,M). We say that ν is absolutely continuous with respect to µ, andwrite ν << µ, if |ν| << µ, as positive measures, that is, for every E ∈M thefollowing implication holds:

µ(E) = 0 ⇒ |ν|(E) = 0 .

(ii) If µ or ν are signed Rm-valued signed measures on measure space (X,M),

we say that they are mutually singular, and write µ ⊥ ν, if |µ| and |ν| aremutually singular, as positive measures, that is, there exists E ∈M such that|µ|(E) = |ν|(X \ E) = 0.

Remark 1.51. Observe that, given a positive measure µ on a measure space (X,M),then vector measure µw defined in (1.19) is trivially absolutely continuous with respectto µ by Proposition 1.47.

Lebesgue decomposition theorem for vector signed measures 1.52. Let νand µ be respectively a Rm

-valued σ-finite measure and a σ-finite positive measure ona measure space (X,M). Then there is a decomposition of ν such that

(1.21) ν = νac + νs ,

where νac and νs are still Rm-valued signed measures on (X,M) with νac << µ and

νs ⊥ µ. The decomposition is unique. Moreover there exists a unique vector functionw = (w1, . . . , wm) : X → Rm

with either wi,+ or wi,− (i = 1, . . . ,m) integrablefunctions w.r.t. µ such that

(1.22) νac(E) = µw(E) =

∫E

w dµ ∀E ∈M .

Proof. Let ν = (ν1, . . . , νm) : M → Rm. By definition, for each i = 1, . . . ,m,νi : M → R is a σ-finite signed vector measure. By Theorem 1.36, there are twosigned measures ν

i,ac, νi,s : M→ R such that

(1.23) νi,ac << µ, νi,s ⊥ µ, νi = νi,ac + νi,s ,

and there exists a measurable function wi : X → R such that either wi,+ or wi,− isintegrable with respect to µ such that

(1.24) νi,ac(E) =

∫E

wi dµ ∀E ∈M .

Moreover both decomposition (1.23) and representation (1.24) are unique.Define the signed vector measures νac, νs :M→ Rm

νac := (ν1,ac, . . . , νm,ac) and νs := (ν1,s, . . . , νm,s)

and the vector function w := (w1, . . . , wm) : X → Rm. Then it is trivial to see that

(1.21) and (1.22) now hold for the signed vector measure ν. Therefore the proof isaccomplished.

Each signed vector measure ν is trivially absolutely continuous with respect to itstotal variation |ν|. The following useful decomposition for vector measures immedi-ately follows from the Lebesgue decomposition theorem for signed vector measures1.52, Proposition 1.47 and Remark 1.38.

21

Corollary 1.53 (Polar decomposition for vector measures). Let ν be a Rm -valuedmeasure on the measure space (X,M). Then there exists a unique measurable vectorfunction wν : X → Rm

with |wν(x)| = 1 |ν| a.e. x ∈ X such that ν = |ν|wν , that is

ν(E) =

∫E

wν d|ν| ∀E ∈M .

.

Proof. Let us first notice that, by Theorem 1.42 (ii), both ν and |ν| are finite andν << |ν|. By Theorem 1.52 and Remark 1.38 there exists a vector function wν :X → Rm

such that

(1.25) ν(E) =

∫E

wν d |ν| ∀E ∈M .

By (1.25) and Proposition 1.47 , we can infer that

|ν|(E) =

∫E

|wν | d |ν| ∀E ∈M .

Since |ν| is finite, we have that |wν(x)| = 1 |ν|-a.e. x ∈ X and the proof is accom-plished.

1.5. Spaces Lp(X,µ) and their main properties. Riesz representation theo-rem.

Completeness and dual space of Lp(X,µ)

In this subsection we will only request that (X,M, µ) is a measure space.Let us introducel the space of p-integrable functions with respect to measure µ.

Definition 1.54. Let p ∈ [1,∞],

Lp(X,µ) :=f : X → R : f is measurable and ‖f‖Lp < +∞

where

‖f‖Lp = ‖f‖Lp(X,µ) :=

(∫

X

|f(x)|p dµ(x)

)1/p

if 1 6 p < ∞

inf M > 0 : |f(x)| 6 M µ− a.e. x ∈ X if p =∞.

The quantity ‖f‖Lp is called the Lp norm of f on measure space (X,M, µ). WhenX = Ω is an open subset of Rn, µ = Ln, M = Ω ∩Mn, where Mn denotes the classof n-dimensional Lebesgue measurable sets of Rn and d the Euclidean distance, wewill simply denote Lp(X,µ) as Lp(Ω).

Remark 1.55. When dealing with measure- theoretic or functional-analytic prope-nies of functions and Lp spaces, it is often convenient to consider functions that agreea.e. as identical, thinking of the elements of Lp spaces as equivalence classes; in par-ticular, this makes ‖ · ‖Lpa norm. We shall follow this path whenever our statementswill depend only on the equivalence class without further mention, provided that thisis clear from the context.

Let us recall the following fundamental result concerning the completeness of Lp

(see [GZ, Theorem 6.24]).

22

Theorem 1.56 (Fisher-Riesz,1907). (Lp(X,µ), ‖ · ‖Lp) is a B.s. if 1 6 p 6 ∞.Moreover L2(X,µ) turns out to be a Hilbert space with respect to the scalar product

(f, g)L2 :=

∫X

f g dµ f, g ∈ L2(X,µ) .

As a consequence of the proof of Riesz- Fisher’s Theorem we have the followinguseful result.

Theorem 1.57. Let (fh)h ⊂ Lp(X,µ) and f ∈ Lp(X,µ) with 1 6 p 6 ∞. Supposethat

(MC) limh→∞‖fh − f‖Lp(X,µ) = 0 .

Then, there exist a subsequence (fhk)k and a function g ∈ Lp(X,µ) such that

(i) fhk(x)→ f(x) µ− a.e. x ∈ X;(ii) |fhk(x)| 6 g(x) µ− a.e. x ∈ X, ∀ k.


Remark 1.58. The implication (MC)⇒ fh(x)→ f(x) µ-a.e. x ∈ Ω may not hold.

Historical notes: ([P, Sections 1.1.4,1.5.2,4.4.1]) Fisher and Riesz invented theHilbert space L2([a, b]) in 1907, by proving its completeness. Both authors observedthe significance of Lebesgue’s integral as the basic ingredient. Subsequently, in 1909,Riesz extended this definition to exponents 1 < p < ∞ and described how theinterval [a, b] can be replaced by any measurable set of Rn. Lp spaces are sometimescalled Lebesgue spaces, named after H. Lebesgue (Dunford & Schwartz 1958, III.3),although according to Bourbaki (1987) they were first introduced by Riesz.

Definition 1.59. Let 1 6 p < ∞ and denote

p′ :=

p

p− 1if 1 < p < ∞

∞ if p = 1

p′ is called conjugate exponent of p.

Theorem 1.60 (Holder inequality). Let p and p′ be conjugate exponents, 1 6 p < ∞Let f ∈ Lp(X,µ) and g ∈ Lp′(X,µ). Then f g ∈ L1(X,µ) and

‖fg‖L1(X,µ) 6 ‖f‖Lp(X,µ) ‖g‖Lp′ (X,µ)

Proof. See [R1, Theorem 3.8] or [GZ, Theorem 6.20].

The Holder inequality establishes a duality between Lp(X,µ) and the dual spaceof Lp(X,µ), denoted (Lp(X,µ))′ according to the notation of functional analysis: ifu ∈ Lp′(X,µ), it is well defined the continuous linear functional T (u) : Lp(X,µ)→ R,that is T (u) ∈ (Lp(X,µ))′, by

T (u)(f) := 〈T (u), f〉(Lp(X,µ))′×Lp(X,µ) :=

∫X

u f dµ ∀f ∈ Lp(X,µ) .

The question naturally arises whether all continuous linear functionals on Lp(X,µ)have this form, and whether the representation is unique. The answer is affirmativeif 1 < p < ∞. It is also affirmative if p = 1, provided that an additional conditionon measure µ.

23

Riesz representation theorem for the dual space of Lp 1.61. If 1 < p < ∞,then the mapping T : Lp

′(X,µ)→ (Lp(X,µ))′, defined by

〈T (u), f〉(Lp(X,µ))′×Lp(X,µ) :=

∫X

u f dµ ∀f ∈ Lp(X,µ) ,

is an isometric isomorphism, that is, T is a linear, one-to-one, onto mapping and

‖T (u)‖(Lp(X,µ))′ = ‖u‖Lp′ (X,µ) ∀u ∈ Lp′(X,µ) .

If p = 1, the same conclusion holds under the additional assumption that µ isσ-finite. We will mean this feature by means of the identification

(1.26) Lp′(X,µ) ≡ (Lp(X,µ))′ .

Proof. See [GZ, Theorem 6.43] or [F, Theorem 6.15], and [R1, Theorem 6.16]) if p = 1provided µ is σ-finite.

Remark 1.62. Identification (1.26) may fail in the other cases: see [F, section 6.2].

Historical notes:([P, Section 2.2.7]) The identity (Lp(a, b))′ = Lp′(a, b) with 1 <

p < ∞ was proved by F. Riesz in 1909. The limit case p = 1 is due to Steinhuas in1919.

Density of continuous functions in (Lp(X,µ), ‖ · ‖Lp). Riesz representation theorem.

The subject of this subsection concerns measure and integration theory on locallycompact metric spaces. We have seen that the Lebesgue measure on Rn interactsnicely with the topology on Rn - measurable sets can be approximated by open orcompact sets, and integrable functions can be approximated by continuous functions- and it is of interest to study measures having similar properties on more generalspaces. Moreover, it turns out that certain linear functionals on spaces of continuousfunctions are given by integration against such measures. This fact constitutes animportant link between measure theory and functional analysis, and it also providesa powerful tool for constructing measures.

In this subsection we will only request that (X,B(X), µ) is a measure space with(X, d) locally compact, separable metric space (we will often use the abbrevia-tion l.c.s. in the following) and µ a Radon measure.

Let us begin to deal with the approximation of continuous functions in Lp. Notethat, under the above assumptions, C0

c(X) ⊂ Lp(X,µ) for each p ∈ [1,∞], providedthat µ is a Radon measure on X.

Theorem 1.63 (Approximation in Lp by continuous functions ). Let (X,B(X), µ)be a measure space with (X, d) l.c.s. and µ Radon measure. Then C0

c (X) is dense in(Lp(X), ‖ · ‖Lp), provided that 1 6 p < ∞.

Proof of Theorem 1.63. The proof can be carried out as in the case of Lp(Ω), bymeans of the approximation by simple functions ( Theorem 1.20) and Lusin’s theorem(Theorem 1.23): see [R1, Theorem 3.14] and also [SC, Theorem 2.59] .

Remark 1.64. Assume that f ∈ L∞(X,µ) ∩ Lp(X,µ) (1 6 p < ∞). Then byTheorems 1.63 and 1.57, it follows that there exists a sequence (fh)h ⊂ C0

c(X) such

24

that

fh → f in Lp(X,µ) as h→∞ and |fh(x)| 6 ‖f‖L∞(X,µ) ∀x ∈ X .

Indeed, by Theorems 1.63 and 1.57, there exist a sequence (gh)h ⊂ C0c(X) and a

function g ∈ Lp(X,µ) such that

gh → f in Lp(X,µ) and |gh(x)| 6 g(x) µ-a.e. x ∈ X .

. Then, let define fh : X → R as

fh(x) = minmaxgh(x),−‖f‖L∞(X,µ), ‖f‖L∞(X,µ)

We are now going to establish an important relationship between Radon measuresand suitable linear bounded (or continuous) functionals on the space of compactlysupported continuous functions, called Riesz representation theorem and due to F.Riesz. This relationship will turn out to be a fundamental bridge between measuretheory and functional analysis.

Let us first recalls some preliminary topological results.

Urysohn’s lemma 1.65 (1925). Let X be a locally compact metric space, let K ⊂ Xand V ⊂ X be, respectively, a compact set and an open set such that K ⊂ V . Thenthere exists a function ϕ ∈ C0

c(X) such that

0 6 ϕ 6 1, ϕ ≡ 1 in K and spt(ϕ) ⊂ V .

Proof. We skip the proof (see [GZ, Lemma 9.7]).

Lemma 1.66 (Partition of unity). Let (X, d) be a locally compact metric space.Assume that V1, . . . , VN and K are respectively open sets and a compact set in Xsuch that

K ⊂ ∪Ni=1Vi .

Then there exist fi ∈ C0c(X) (i = 1, . . . , N) such that

spt(fi) ⊂ Vi, 0 6 fi 6 1 ∀ i = 1, . . . , N ;

N∑i=1

fi(x) = 1 ∀x ∈ K .

A family of functions f1, . . . , fN satisfying previous properties is called a partitionof unity subordinate to the open covering V1, . . . , VN of K.

Proof. See [GZ, Theorem 9.8] or [R1, Theorem 2.13].

Let µ be a Radon measure on measure space (X,B(X)). We can associate to µ alinear functional Lµ : C0

c(X)→ R defined by

(1.27) Lµ(u) :=

∫X

u dµ ∀u ∈ C0c(X) .

Notice that Lµ is positive . Let us recall that

Definition 1.67. A linear functional L : C0c(X) → R is said to be positive (or also

monotone) if L(u) > 0 whenever u > 0 (or, equivalently, L(u) > L(v) wheneveru > v).

25

In this definition there is no mention of continuity, but it is worth noting thatpositivity itself implies a rather strong continuity property. More precisely, let usendow (C0

c(X))m by the ∞- (or also uniform) norm defined by

(1.28) ‖u‖∞ := supx∈X|u(x)|Rm if u ∈ C0

c(X) .

For the sake of simplicity, we will denote Rm-norm simply by | · | from now on.

Proposition 1.68. If L : C0c(X)→ R is a positive linear functional, for each compact

K ⊂ X there is a positive constant CK (depending on K) such that

(1.29) sup|L(u)| : u ∈ C0

c(X), ‖u‖∞ 6 1, spt(u) ⊂ K6 CK .

Proof. See [F, Proposition 7.1].

We will assume inequality (1.29) as definition of continuity (or boundedness) forlinear functionals on the compactly supported continuous functions, even in the vectorcase L : (C0

c(X))m → R.

Definition 1.69. A linear functional L : (C0c(X))m → R is said to be continuous (or

bounded) if, for each compact set K ⊂ X, there is a positive constant CK (dependingon K) such that (1.29) holds.

Remark 1.70. For the sake of simplicity, assume that m = 1. Then it can beproved that the notion of continuity in Definition 1.69 is induced by a topology τ onC0c(X) for which (C0

c(X), τ) turns out to be a locally convex , complete topologicalvector space, which is not metrizable (see [T]). Indeed let (Ah)h be an increasingsequence of relative compact open sets of (X, d) . Then, if we denote by C0

0(Ah)the space of continuous function vanishing at infinity on Ah (see Definition 1.80),it can be shown that (C0

0(Ah), ‖ · ‖∞) is a Banach space (see Proposition 1.81) andC0c(X) = ∪∞h=1C

00(Ah) . If ih : C0

0(Ah)→ C0c(X) (h = 1, 2, . . . ) denotes the inclusion

map, the topology τ turns out to be the strongest topology on C0c(X) for which maps

ih : (C00(Ah), ‖ · ‖∞) → (C0

c(X), τ) are continuous for each h. The existence of sucha topology τ can be provided meaning C0

c(X) as the vector topological space, alsocalled LF-space, countable inductive strict limit of Banach spaces (C0

0(Ah), ‖ · ‖∞)h(see, for instance, [T, Chap. XIII]).

Moreover a notion of convergence can be induced by means of this topology.Exercise: A linear functional L : (C0

c(X))m → R is continuous (according toDefinition 1.69) if and only if limh→∞ L(uh) = L(u) for each (uh)h, u in (C0

c(X))m

satisfying(1.30)

uh → u uniformly on X and there exists a compactK ⊂ X, sptu ∪∞⋃h=1

sptuh ⊂ K .

Definition 1.71. We write that uh → u in (C0c(X))m, if (1.30) holds.

Example 1.72. If µ is a Radon measure on X and w = (w1, . . . , wm) : X → Sm−1 :=y ∈ Rm : |y| = 1 is a (Borel) measurable function, we may define a continuouslinear functional wµ : (C0

c(X))m → R setting

(1.31) wµ(u) :=

∫X

(w, u)Rm dµ =

∫X

m∑i=1

wi ui dµ u = (u1, . . . , um) ∈ (C0c(X))m ,

26

which is trivially an extension of the functional defined in (1.27) when m = 1. Wewill see that each continuous linear functional L : (C0

c(X))m → R can be representedby form (1.31) for suitable w and µ.

Riesz representation theorem 1.73. Let (X, d) be a separable, locally compactmetric space and let L : (C0

c(X))m → R be a continuous linear functional. Thenthere exist a Radon measure µL : B(X) → [0,∞] and a Borel measurable vectorfunction wL : X → Sm−1 such that

(1.32) L(u) =

∫X

(wL, u)Rm dµL ∀u ∈ (C0c(X))m ,

that is, L = wLµ, and µL is characterized by the following identity: for each open setA ⊂ X

(1.33) µL(A) = sup L(u) : u ∈ (Cc(X))m, sptu ⊂ A, ‖u‖∞ 6 1 .Moreover representation (1.32) is unique.

Definition 1.74. Let (X, d) be a locally compact metric space and consider themeasure space (X,B(X)). Let us also denote by Bcomp(X) the class of Borel setswhich are relatively compact in X.

(i) A Rm-valued Radon vector measure ν on X is a set function ν : Bcomp(X)→Rm for which there exist a Borel vector function wν : X → Sm−1 and apositive Radon measure µ : B(X)→ [0,∞] such that

(1.34) ν(E) =

∫E

wν dµ ∀E ∈ Bcomp(X) .

(ii) A Rm-valued finite Radon vector measure is a set function ν : B(X)→ Rm forwhich there exist a Borel vector function wν : X → Sm−1 and a positive finiteRadon measure µ : B(X)→ [0,∞] such that (1.34) holds for all E ∈ B(X).

We will write that ν = wν µ in both previous cases.We denote by (Mloc(X))m (respectively (M(X))m) the space of Rm-valued Radon

(resp. Rm-valued finite Radon ) measures on X.

Remark 1.75. Observe that, by Corollary 1.53, we could equivalently define a Rm-valued Radon vector measure (respectively a Rm-valued finite Radon vector measure)as a set function ν : Bcomp(X) → Rm such that, for each compact K ⊂ X, ν :B(K)→ Rm is a vector measure (respectively as a vector measure ν : B(X)→ Rm)

Remark 1.76. It easy to see that (Mloc(X))m turns out to be vector space.

Before the proof of the Riesz representation theorem we need some technical lemma.

Lemma 1.77. Let L : (C0c(X))m → R be a continuous linear functional. Let µ∗L :

P(X)→ [0,∞] be the set function defined as

(1.35) µ∗L(E) := inf µL(A) : A open, A ⊇ E if E ⊂ X

where µL(A) is the quantity in (1.33). Then µ∗L is a Radon outer measure on X.

Proof. Denote ν = µ∗L and µ = µL. Let us first observe that, if U is open, then

(1.36) µ(U) = ν(U) ,

27

that is the definition of ν is consistent. Indeed we immediately get that

µ(U) > ν(U) .

If A is open and A ⊇ U , then by definition (see (1.33)), we get

µ(U) 6 µ(A) ,

which implies

µ(U) 6 ν(U) .

Thus (1.36) follows. Let us now divide the proof in three steps.1st step. We prove that ν is an outer measure.Let us first show that ν is countably subaddtive on open sets, that is, if (Ah)h is a

sequence of open sets and A = ∪∞h=1Ah, then

(1.37) ν(A) 6∞∑h=1

ν(Ah) .

Let u ∈ (C0c(A))m with ‖u‖∞ 6 1 and let K := spt(ϕ). Since K ⊂ A is a compact

set, there exists an integer N such that K ⊂ ∪Nh=1Ah. Let us consider a partition ofunity ϕ1, . . . , ϕN subordinate to A1, . . . , AN and K, that is

ϕh ∈ C0c(Ah), 0 6 ϕh 6 1,

N∑h=1

ϕh(x) = 1 ∀x ∈ K .

Since u =∑N

h=1 uϕh and uϕh ∈ (C0c(Ah))

m with |uϕh| 6 1

L(u) =N∑h=1

L(uϕh) 6N∑h=1

ν(Ah) 6∞∑h=1

ν(Ah) .

Then (1.37) follows passing to the supremum over all u ∈ (C0c(A))m with ‖u‖∞ 6 1.

Let us now prove that ν is countably subadditive, that is

(1.38) ν(E) 6∞∑h=1

ν(Eh) if E ⊂ ∪∞h=1Eh .

Let us first observe that ν is non decreasing w.r. t. the inclusion, that is

ν(E) 6 ν(F ) if E ⊂ F .

Without loss of generality we can suppose ν(Eh) <∞ for each h. For each ε > 0 andh there is an open set Ah ⊇ Eh such that

ν(Ah) < ν(Eh) +ε

2h.

Thus, by (1.37),

ν(E) 6 ν(∪∞h=1Ah) 6∞∑h=1

ν(Ah) 6∞∑h=1

ν(Eh) + ε ∀ ε > 0 .

Therefore (1.38) follows.

28

2nd step. Let us prove that ν is a Borel regular outer measure. In order to provethat ν is a Borel outer measure, by Caratheodory’s criterion (see Theorem 1.5 (ii)),we have only to prove that

(1.39) ν(E ∪ F ) > ν(E) + ν(F ) whenever d(E,F ) > 0 .

Assume E, F are open, and let ϕ ∈ (C0c(E ∪ F ))m with |ϕ| 6 1. Then, since E and

F are disjoint,

ϕ = ϕ|E + ϕ|F , ϕ|E ∈ (C0c(E))m, ϕ|F ∈ (C0

c(F ))m, |ϕ|E| 6 1, |ϕ|F | 6 1 .

ThusL(ϕ) = L(ϕ|E) + L(ϕ|F ) > ν(E) + ν(F ) ,

and (1.39) follows. In the general case, since 0 < d(E,F ) = d(E, F ), there exist opensets A1, A2 such that E ⊂ A1 and F ⊂ A2 with d(A1, A2) > 0. If A is open andE∪F ⊂ A, then d(A1∩A,A2∩A) > 0 and E ⊂ A1∩A, F ⊂ A2∩A, so that (1.39)onopen sets implies

ν(A) > ν ((A1 ∩ A) ∪ (A2 ∩ A)) > ν(A1 ∩ A) + ν(A2 ∩ A) > ν(E) + ν(F ) .

As A is arbitrary, (1.39) follows. Hence ν is a Borel measure. Moreover ν is Borelregular, since, if E ⊂ X, ν(E) < ∞ and (Ah)h are open sets with E ⊂ Ah andlimh→∞ ν(Ah) = ν(E), then B := ∩∞h=1Ah is a Borel set with E ⊂ B and ν(E) =ν(B). If ν(E) = ∞, we can choose as Borel envelope B = X.

3rd step. Let us prove that ν is finite on compact sets. Let us recall, that byLemma 1.17, there exists an increasing sequence of open sets (Vi)i such that

X = ∪∞i=1Vi, Vi compact for each i .

In particular, since L is bounded, by (1.33) and (1.29), it follows that

ν(Vi) = µ(Vi) < ∞ ∀ i .Given a compact set K ⊂ X, then there exists an integer i0 such that

K ⊂ ∪i0i=1Vi .

Therefore, since ν is subadditive,

ν(K) 6i0∑i=1

ν(Vi) < ∞ ,

and the proof is accomplished.

Proof of the Riesz representation theorem 1.73. By Lemma 1.77, ν := µ∗L is a Radon

outer measure on X. Let us now define functional L : C0c(X; [0,∞))→ [0,∞) as

L(ϕ) := supL(u) : u ∈ (C0

c(X))m, |u| 6 ϕ, ϕ ∈ C0c(X; [0,∞))

.

We will divide the proof in three steps. In step one we will show that L is additive,positively homogeneous of degree one, and monotone on C0

c(X; [0,∞)). In step two,we will show the inequality

(1.40) L(ϕ) 6∫X

ϕdν ∀ϕ ∈ C0c(X; [0,∞)) .

Finally, in step three, by using the Riesz representation theorem for the dual ofL1(X,µ) (see Theorem 1.61) we can conclude the proof.

29

1st step. We show that, whenever ϕi ∈ C0c(X; [0,∞)) (i = 1, 2) and c > 0, we

have

(1.41) L(ϕ1 + ϕ2) = L(ϕ1) + L(ϕ2) ,

(1.42) L(c ϕ1) = c L(ϕ1),

(1.43) L(ϕ1) 6 L(ϕ2), if ϕ1 6 ϕ2 .

It si immediate that (1.43) follows by the definition of L. let us prove the remainingproperties. By definition of L and the linearity of L, for each ui ∈ (C0

c(X))m (i = 1, 2)with |ui| 6 ϕi, c > 0 , we can infer

L(ϕ1 + ϕ2) > L(u1 + u2) = L(u1) + L(u2) ,

L(c ϕ1) > L(c u1) = c L(u1) ,

L(c ψ1) = c L(ψ1) 6 c L(ϕ1) ,

L(ui) 6 L(ϕi) i = 1, 2, .

Then, by the previous inequalities, it follows respectively that

(1.44) L(ϕ1 + ϕ2) > L(ϕ1) + L(ϕ2) ,

(1.45) L(c ϕ1) > c L(ϕ1) ,

(1.46) L(c ϕ1) 6 c L(ϕ1) .

Therefore, by (1.45) and (1.46) we have proven (1.42), and we have only to provethe inverse inequality of (1.44) for showing (1.41) . Now let u ∈ (C0

c(X))m be suchthat |u| 6 ϕ1 + ϕ2, and set

ui :=

ϕi

ϕ1 + ϕ2

u on ϕ1 + ϕ2 > 0

0 otherwisei = 1, 2 .

Exercise: Prove that ui ∈ (C0c(X))m, |ui| 6 ϕi (i = 1, 2) and u = u1 + u2.

ThereforeL(u) = L(u1) + L(u2) 6 L(ϕ1) + L(ϕ2),

and complete the proof by the arbitrariness of u.2nd step. Let us now prove (1.40). Given ϕ ∈ C0

c(X; [0,∞)) and ε > 0, lett0, . . . , tN be real numbers such that(1.47)t0 < 0 < t1 < · · · < tN−1 < sup

Xϕ < tN , th+1 − th 6 ε with h = 1, . . . , N − 1

and consider the partition E1, . . . , EN of spt(ϕ) by disjoint Borel sets, defined as

Eh = x ∈ spt(ϕ) : th−1 < ϕ(x) 6 th , 1 6 h 6 N .

Since ν is a Radon o.m., by Theorem 1.14, there exist open sets Ah with Eh ⊂ Ahand

(1.48) ν(Ah) 6 ν(Eh) +ε

N, 1 6 h 6 N .

30

If necessary replacing Ah with x ∈ Ah : ϕ(x) < th + ε, we can also assume

(1.49) ϕ < th + ε on Ah .

Finally, let f1, . . . , fN be a partition of unity subordinated to the open coveringA1, . . . , AN of the compact set spt(ϕ), namely fh ∈ C0

c(Ah) ,0 6 fh 6 1, and∑Nh=1 fh(x) = 1 on spt(ϕ). Since ϕ =

∑Nh=1 fh ϕ , by step 1 and (1.49), we find that

(1.50) L(ϕ) =N∑h=1

L(fhϕ) 6N∑h=1

(th + ε) L(fh) .

If ψ ∈ (C0c(X))m, and |ψ| 6 fh, then spt(ψ) ⊂ Ah and |ψ| 6 1. Hence, L(fh) 6

ν(Ah) and, by (1.47) and (1.48), we find that

L(ϕ) 6N∑h=1

(th + ε)(ν(Eh) +

ε

N

)6

N∑h=1

(th−1 + 2ε)(ν(Eh) +

ε

N

)=

N∑h=1

th−1 ν(Eh) +ε

N

N∑h=1

th−1 + 2εN∑h=1

ν(Eh) + 2ε2

6∫X

ϕdν + tNε+ 2ε ν(spt(ϕ)) + 2ε2

6∫X

ϕdν + ε

(supXϕ+ ε+ 2 ν(spt(ϕ)) + 2ε

)If we let ε→ 0+ in the previous inequality, (1.40) follows.

3rd step. Given e ∈ Sm−1, we define Le : C0c(X)→ R by

Le(ϕ) := L(ϕ e) ϕ ∈ C0c(X) .

By (1.40), we find that, for every ϕ ∈ C0c(X),

Le(ϕ) 6 supL(ψ) : ψ ∈ (C0

c(X))m, |ψ| 6 |ϕ|

= L(|ϕ|) 6∫X

|ϕ| dν .

By the approximation in Lp by continuous functions (see Theorem 1.63), we mayextend Le as a linear functional Le : L1(X, ν)→ R such that

|Le(u)| 6∫X

|u| dν = ‖u‖L1(X,µ) ∀u ∈ L1(X, ν) .

Thus, by the Riesz representation theorem for the dual of L1(X, ν), there existswe ∈ L∞(X, ν) such that

L(u e) =

∫X

uwe dν ∀u ∈ L1(X, ν) .

31

If we set wL := (w1, . . . , wm) : X → Rm, wi := wei , where e1, . . . , emdenotes thestandard basis of Rm, then wL is bounded and ν-measurable, with

(1.51) L(ϕ) =m∑i=1

Lei(ϕi) =m∑i=1

∫X

wi ϕi dν =

∫X

(wL, ϕ)Rm dν

for every ϕ = (ϕ1, . . . , ϕm) ∈ (C0c(X))m. Let us now prove that

(1.52) |wL(x)| = 1 ν-a.e. x ∈ X .

By (1.36) and (1.51), it follows that, for each bounded open set A ⊂ X,

ν(A) = sup L(ϕ) : ϕ ∈ (Cc(X))m, sptϕ ⊂ A, |ϕ| 6 1

= sup

∫X

(wL, ϕ)Rm dν : ϕ ∈ (Cc(X))m, sptϕ ⊂ A, |ϕ| 6 1

6∫A

|wL| dν .

(1.53)

From (1.53), it follows that

|wL| > 0-a.e. in X and u := χ|wL|>0wL|wL|

∈ (L1(A, ν))m .

By Remark 1.64, there exists a sequence (ϕh)h ⊂ (C0c(A))m such that

ϕh → u in (L1(A, ν))m and |ϕh| 6 1 .

This implies that

(1.54) (wL, ϕh)Rm → |wL| in L1(A, ν)

Therefore, since

ν(A) >∫A

(wL, ϕh)Rm dν ,

by (1.54) and passing to the limit, as h → ∞, in the previous inequality, it followsthat

(1.55) ν(A) >∫A

|wL| dµ .

Thus, by (1.53) and (1.55), we can infer that

ν(A) =

∫A

|wL| dν for each bounded open set A ⊂ X

which implies (1.52).Let us prove that we can choose wL : X → Sm−1 as a Borel measurable vector

function. Indeed, observe that, by Corollary 1.24, we can assume that wL : X → Rm

is Borel measurable and satisfying (1.51). On the other hand, by (1.52), there existsa ν-null set N ⊂ X (which, a priori, could be not a Borel set) such that |wL(x)| = 1for each x ∈ X \N . Since ν is Borel regular, there exists a Borel ν-null set B ⊃ N .Thus, by changing the value of wL on B, for instance putting wL := e1 on B, we getthat wL : X → Sm−1 is still Borel measurable and satisfies (1.51). The uniqueness ofν and wL (ν-a-e.) follows in a standard way.

32

Finally, without loss of generality, since ν(A) = µ∗L(A) = µL(A) for each open setA ⊂ X, we still denote µL the outer measure defined in (1.35) and yield a Radonmeasure µL : B(X)→ [0,∞] satisfying the desired properties.

Riesz representation theorem 1.73 provides a characterization of the measuresspacesM(X)m and (Mloc(X))m as dual spaces of suitable spaces of continuous func-tions.

Observe first that functional (1.27) can make sense even for Radon signed measures.Indeed, if ν ∈ Mloc(X), that is, ν = wν µ with µ positive Radon measure on X andwν : X → −1, 1 Borel function, then , according also to Definition 1.48 (i), it iswell defined

(1.56) Lν(u) :=

∫X

u dν =

∫X

wν u dµ ∀u ∈ C0c(X) .

The functional Lν : C0c(X) → R still turns out to be a linear continuous functional

according to Definition 1.69. Thus a trivial consequence of Theorem 1.73 is thefollowing

Corollary 1.78 (Characterization ofMloc(X)). Let (X, d) be a locally compact sep-arable metric space and define(1.57)(C0

c (X))′ :=L : C0

c (X)→ R : L is linear and continuous w.r.t. Definition 1.69.

Let us define the map

(1.58) I : Mloc(X)→ (C0c (X))′ I(ν) := Lν .

Then I is an isomorphism (between vector spaces).

If ν ∈ M(X) functional (1.56) is still well defined since M(X) ⊂ Mloc(X). Inthis case the functional is actually an element of the dual of normed vector space(C0

c(X), ‖ · ‖∞) and we will write Lν ∈ (C0c(X), ‖ · ‖∞)′.

Proposition 1.79. Let ν ∈ M(X), and let Lν : C0c(X) → R be the functional in

(1.56). Then

‖Lν‖(C0c(X),‖·‖∞)′ := sup

|Lν(u)| : u ∈ C0

c(X), ‖u‖∞ 6 1

= |ν|(X) .

Proof. It is a special case of (1.33).

Functional Lν can actually be extended to the completion of (C0c(X), ‖ · ‖∞), that

is the class of continuous functions on (X, d) vanishing at infinity.

Definition 1.80 (Functions vanishing at infinity). Let (X, d) be a locallycompact metric space.

(i) A function u : X → R is said to vanish at infinity if for every ε > 0 there is acompact set K ⊂ X such that

|u(x)| < ε ∀x ∈ X \K .

(ii) The class of all continuous u : X → R which vanish at infinity is called C00(X),

33

It is clear that C0c (X) ⊂ C0

0(X) and that the two classes coincide if (X, d) iscompact. More precisely, if (X, d) is compact, then

C0c (X) = C0

0(X) = C0(X) .

It is also well known that

Proposition 1.81. C00(X) is the completion of normed vector space (C0

c(X), ‖ · ‖∞).

Proof. See [R1, Theorem 3.17].

Proposition 1.82. Let ν ∈M(X) and define

(1.59) ‖ν‖ := |ν|(X) .

Then (M(X), ‖ · ‖) is a real normed vector spaces.

Proof. [F, Proposition 7.16].

Theorem 1.83 (Characterization ofM(X)). Let (X, d) be a locally compact separablemetric space. Let I the map in (1.58). Then

(i) I(M(X)) = (C00(X), ‖ · ‖∞)′;

(iI) I : (M(X), ‖ · ‖) → (C00(X), ‖ · ‖∞)′ is a topological isomorphism, that is an

algebraic isomorphism, continuous with its inverse.

Proof. [R1, Theorem 6.19].

Corollary 1.84. Let (X, d) be a compact metric space. Then (C0(X), ‖ · ‖∞)′ isisometrically isomporphic to M(X).

Compactness in (Lp(Ω), ‖ · ‖Lp).In this section we are going to deal with some compactness results in Lp spaces.

We will only state these results, without proofs which can be found in [B, Section4.5].

Let f : Rn → R and v ∈ Rn be given, then we define by τvf : Rn → R thev-translated function of f defined by

(τvf)(x) := f(x+ v) .

Theorem 1.85 (M. Riesz- Frechet- Kolmogorov). Let F be a bounded subset in(Lp(Rn), ‖ · ‖Lp) with 1 6 p < ∞. Suppose that limv→0 ‖τvf − f‖Lp = 0 uniformlyfor f ∈ F , that is

∀ ε > 0∃ δ = δ(ε) > 0 such that ‖τvf − f‖Lp < ε ∀ v ∈ Rn with |v| < δ,

∀ f ∈ F .(ENF)

Then F|Ω := f |Ω : f ∈ F is relatively compact in (Lp(Ω), ‖ · ‖Lp), i.e. its closureis compact in (Lp(Ω), ‖ · ‖Lp), for each open set Ω ⊂ Rn with finite Lebesgue measure.

From Theorem 1.85 it follows the following compactness criterion in (Lp(Ω), ‖·‖Lp).If f : Ω→ R, let us denote by f : Rn → R the function defined as

f(x) :=

f(x) if x ∈ Ω0 otherwise

.

34

Corollary 1.86. Let Ω ⊂ Rn be an open set with finite measure, let F ⊂ Lp(Ω) and

let F := f : f ∈ F. Assume that

(i) F is bounded in (Lp(Ω), ‖ · ‖Lp) with 1 6 p < ∞;

(ii) limv→0 ‖τvf − f‖Lp = 0 uniformly for f ∈ F , that is, F satisfies ENF .

Then F is relatively compact in (Lp(Ω), ‖ · ‖Lp).

Proof. From Theorem 1.85, F is relatively compact. Notice now that F is relativelysequentially compact in (Lp(Rn), ‖ · ‖Lp) if and only if F is relatively sequentiallycompact in (Lp(Ω), ‖ · ‖Lp). Thus, the characterization of compact sets in metricspaces completes the proof.

Eventually recall the following characterization of compactness in (Lp(Rn), ‖ · ‖Lp).

Theorem 1.87. Let F ⊂ Lp(Rn) with 1 6 p < ∞. Then F is relatively compact in(Lp(Rn), ‖ · ‖Lp) if and only if

(i) F is bounded in (Lp(Rn), ‖ · ‖Lp);(ii) for each ε > 0 there exists rε > 0 such that

‖f‖Lp(Rn\B(0,rε)) < ε ∀ f ∈ F ;

(iii) limv→0 ‖τvf − f‖Lp = 0 uniformly for f ∈ F .

Remark 1.88. (i) The assumption (ENF) is necessary in Theorem 1.85. Indeed,consider the family F := fh : h ∈ N where fh : R → R is defined as fh(x) :=h if 0 6 x 6

1

h0 otherwise

and let Ω := (0, 1). Then it is easy to see that ‖fh‖L1(R) = 1

for each h ∈ N and F|Ω is not relatively compact in (L1(Ω), ‖ ·‖L1), since there are nosubsequences of (fh)h converging in L1(Ω) (see Exercise III.6). On the other hand,for given v > 0, for each h > 1

v

‖τvfh − fh‖L1(R) >∫ 0

−∞fh(x+ v) dx =

∫ v

0

fh(x) = 1 .

Thus, (ENF) does not hold for F .(ii) If Ω has not finite measure, then the conclusions of Theorem 1.85 need not hold.Indeed , consider the family F := fh : h ∈ N where fh : R → R is definedas fh(x) := f(x + h) where f ∈ Lip(R) with spt(f) = [−a, a], a > 0, and f notidentically vanishing. Then

(1.60) ‖fh‖L1(R) = ‖f‖L1(R) > 0 ∀h .

Moreover F satisfies (ENF), because

|τvf(x)− f(x)| = |f(x+ v)− f(x)| 6 L |v|χ[−a−1,a+1](x) ∀x ∈ R, v ∈ [−1, 1] ,

and

‖τvfh − fh‖L1(R) = ‖τvf − f‖L1(R) ∀hwhere L := Lip(f). Let Ω := R and observe now that F = F|Ω is not relativelycompact in (L1(R), ‖·‖L1). Otherwise a contradiction arises by (1.60), since fh(x)→ 0for each x ∈ R,

35

Historical notes:[HOH] A first compactness type-result was proved by Frechet in1908 in the setting of l2. In 1931, Kolmogorov proved the first result in this direction.The result characterizes the compactness in Lp(Rn) for 1 < p <∞ , in the case whereall functions are supported in a common bounded set. Condition (iii) of Theorem1.87 is replaced by the uniform convergence in Lp norm of spherical means of eachfunction in the class to the function itself. (Clearly, our condition (ii) is automatic inthis case.) Just a year later, Tamarkin expanded this result to the case of unboundedsupports by adding condition (ii) of Theorem 1.87. In 1933, Tulajkov expanded theKolmogorov-Tamarkin result to the case p = 1. In the same year, and probablyindependently, Riesz proved the result for 1 < p < ∞, essentially in the form ofour Theorem 1.87. Thus we feel somewhat justified in using the names Kolmogorovand Riesz in referring to the theorem, though we are perhaps being a bit unfair toTamarkin and Tulajkov in doing so. In 1937, M.Frechet replaced conditions (i) and(ii) of Theorem 1.87 with a single condition (”equisummability”), and generalized thetheorem to arbitrary positive p.

1.6. Operations on measures. In this section we discuss some useful and funda-mental operations on measures and related notions: among them, we describe theproduct measures, and state the related Fubini and Tonelli theorems, together withsome consequences.

Definition 1.89 (Support). Let µ be a positive measure on a separable metric spaceX; we call the closed set of all points x ∈ X such that µ(U) > 0 for every neighbour-hood U of x the support of µ denoted spt(µ). In other words,

sptµ = X \ ∪ V : V open, µ(V ) = 0= X \ x ∈ X : ∃ r > 0 such that µ(B(x, r)) = 0

(1.61)

If ν is a signed or signed vector measure, we call the support of ν the support of|ν|.

In the general case of a measure on a measure space (X,M), we say that ν isconcentrated on S ⊂ X if S ∈ M and |ν|(X \ S) = 0. Notice that it is impossiblein general to define a ”minimal” set where a measure is concentrated, hence the setS is not uniquely determined. However, for any pair ν1 and ν2 of mutually singularmeasures there exist pairwise disjoint M-measurable sets S1 and S2 such that νi isconcentrated on Si (i = 1, 2). A property which is not shared by the support.

Example: Consider, for instance, the measure space (R,B(R)) and measures µ1 :=L1 and µ2 :=

∑h 2−h δxh with (xh)h dense in R. Then spt(µ1) = spt(µ2) = R. Even

if measure µ2 is concentrated on the elements of the sequence, but spt(µ1) containsevery accumulation point of the sequence.

Remark 1.90. Notice also that, if X is a separable metric space and ν is a Borelmeasure on X, then spt(ν) is the smallest closed set where ν is concentrated.

Definition 1.91 (Restriction). Let ν be an outer measure X or a positive or vectorsigned measure on the measure space (X,M). Let E ⊂ X.

36

(i) If ν is an outer measure on X, the restricted outer measure of ν to E, writtenν E, is the set function defined by

(1.62) ν E(F ) := ν(E ∩ F ) ∀F ⊂ X .

(ii) If ν is a positive or vector signed measure on the measure space (X,M) andE ∈M, the restricted measure of ν to E, still written ν E, is the set functiondefined in (1.94), but with F ∈M.

Theorem 1.92. (i) If ν is an outer measure on X, then so is ν E. Moreoverevery ν-measurable set is also ν E-measurable.

(ii) If ν is a Borel regular outer measure on X and E is ν-measurable with ν(E) <∞ , then ν E is a Radon outer measure.

(iii) If ν is a positive or vector signed measure on a measure space (X,M) andE ∈M, so is ν E.

Proof. (i) It is clear that ν E is an outer measure. By using the definition ofCaratheodory measurability, the second part of the statement follows, too. Notethat E can be arbitrary here.

(ii) Clearly (ν E)(K) 6 ν(E) < ∞ for each compact set K ⊂ X. Since, byprevious claim (i), every ν-measurable is also (ν E)-measurable, (ν E) is a Borelouter measure. Thus we have only to show that ν E is a Borel regular outer measure.

Since ν is Borel regular, there is a Borel set B such that E ⊂ B and ν(B) = ν(E).Then, since E is ν-measurable and ν(E) < ∞, ν(B \ E) = 0. Let us fix C ⊂ X.Then

(ν E)(C) 6 (ν B)(C) = ν(C ∩B) = ν(C ∩B ∩ E) + ν((C ∩B) \ E)

6 ν(C ∩ E) + ν(B \ E) = (ν E)(C)

Therefore ν B = ν E, so we may as well assume that E is a Borel set.Given C ⊂ X, we must show that there exists a Borel set F such that C ⊂ F and

(ν E)(F ) = (ν E)(C). Since ν is Borel regular, there is a Borel set D such thatC ∩E ⊂ D and ν(C ∩E) = ν(D). Let F := D ∪ (X \E). Since D and E are Borelsets, so is F . Moreover C ⊂ (E ∩ C) ∪ (X \ E) ⊂ F . Finally, since F ∩ E = D ∩ E,

(ν E)(F ) = ν(F ∩ E) = ν(D ∩ E) 6 ν(D)

= ν(C ∩ E) = (ν )E(C) .

Thus (ν E)(F ) = (ν E)(C) and so ν E is Borel regular.(iii) The claim is trivial.

Remark 1.93. By using the same proof of Theorem 1.92 (ii), we can infer that if νis a Borel regular measure and E is a Borel set, then ν E is still Borel regular, evenif ν(E) = ∞.

Theorem 1.92 (ii) allows to generate Radon outer measures by restricting a givenBorel regular outer measure to a measurable set of finite measure. An other interestingprocedure for generating Radon outer measures, in a separable, locally compact metricspace, by restriction of a given Borel regular outer measure is the following.

Theorem 1.94. Let ν be a Borel regular outer measure on a separable, locally compactmetric space (X, d). Let E be a ν-measurable set such that ν E is a locally finiteoutermeasure. Then ν E is a Radon outer measure.

37

Proof. Let us first recall that, from Lemma 1.17, there exists an increasing sequenceof open sets (Vi)i such that

(1.63) X = ∪∞i=1Vi, Vi is compact for each i .

Let ϕ := ν E. Let us first prove that

(1.64) ϕ(K) < ∞ for each compact set K ⊂ X .

If K ⊂ X is a given compact set, by the local fineteness of ϕ, for each x ∈ K thereexists an open ball U(x, rx) such that

(1.65) ϕ(U(x, rx)) < ∞ .

Since K is compact, there is a finite family of open balls U(x1, r1), . . . , U(xm, rm) suchthat

(1.66) K ⊂ ∪mi=1U(xi, ri) .

Thus, by (1.65),(1.66) and the subadditivity of ϕ,(1.64) follows.Since, by Theorem 1.92 (i), every ν-measurable set is also ϕ-measurable, ϕ is a

Borel outer measure. Thus we have only to show that ϕ is a Borel regular outermeasure, that is, given C ⊂ X, we must show that there exists a Borel set F suchthat C ⊂ F and ϕ(F ) = ϕ(C).

Let ϕi := ϕ Vi = ν (E ∩ Vi) (if i ∈ N). Let us observe that

(1.67) ∃ϕ(C) = limi→∞

ϕi(C) for all C ⊂ X .

Indeed, by the continuity of outer measures on increasing sequences of sets and (1.63),

limi→∞

ϕ(C) = limi→∞

(ν (E ∩ C))(Vi) = (ν (E ∩ C))(X) = ϕ(C) .

On the other hand, since ν is Borel regular, E ∩ Vi is ν- measurable, and, by (1.64),ν(E ∩ Vi) 6 ν(Vi) < ∞, by Theorem 1.92 (ii) we can infer that ϕi = ν (E ∩ Vi) is aBorel regular outer measure. Thus, for each i and for a given C ⊂ X, there exists aBorel set Fi such that

(1.68) C ⊂ Fi and ϕi(Fi) = ϕi(C) .

Let F := ∩∞i=1Fi. Then F is still a Borel set with F ⊃ C and

ϕi(C) 6 ϕi(F ) 6 ϕi(Fi) = ϕi(C) .

Thus, it follws that there exists a Borel set F ⊃ C such tha

ϕi(C) = ϕi(F ) ∀ i .

By taking the limit, as i→∞, in the previous identity, we get that

ϕ(C) = ϕ(F ) ,

and, then, the desired conclusion.

Given a measure space (X,M) and a measure on it, we see now how it can becarried on another set Y through a function f : X → Y .

38

Definition 1.95 (Push-forward of a measure, or image measure). Let (X,M) and(Y,N ) be measure spaces, and let f : X → Y be measurable, that is f−1(F ) ∈ Mwhenever F ∈ N . For any positive, real, or vector measure ν on (X,M) we define ameasure f#ν in (Y,N ) by

f#ν(F ) := ν(f−1(F )

)From the previous definition the corresponding change of variable formula for inte-

grals follows immediately: if u is a (reaI- or vector-valued) function on Y integrablewith respect to f#ν, then u f in integrable with respect to ν, and we have theequality:

(1.69)

∫Y

u d(f#ν) =

∫X

u f dν

The very general definition given above can be easily seen to have good propertiesin l.c.s. spaces when f is assumed to be continuous and proper, i.e. such that f−1(K)is compact for any compact K ⊂ Y as the following remark shows.

Remark 1.96. Let X, Y be l.c.s. metric spaces, f : X → Y continuous and proper: the continuity of f ensures that f−1(B) whenever B ∈ B(Y ), and since f is proper,if ν is a Radon measure on X, then f#ν is a Radon measure on Y .

Hausdorff measures provide an important source of examples of Radon measures.We will deeply study them in Chapter III. We are going now to stress some relation-ships between the 1-dimensional Hausdorff measure and the classical notion of lengthmeasure for a curve in Rn.

Example 1.97 (Push-forward of the classical length measure). A set Γ ⊂ Rn is acurve of Rn if there exists a continuous, injective function γ : [a, b] → Rn such thatγ([a, b]) = Γ. The function γ is called a parametrization of Γ. Given a parametrizationγ : [a, b]→ Rn and a subinterval [c, d] ⊆ [a, b], we define the length of γ over [c, d] as

(1.70) length(γ; [c, d]) := sup

N∑i=1

|γ(ti)− γ(ti−1| : t0 = c < t1 < · · · < tN = d

where the supremum is taken over all finite partitions t0 = c < t1 < · · · < tN = dof [c, d]. It can be proved that, if Γ = γ1([a1, b1]) = γ2([a2, b2]) for two givenparametrizations γi : [ai, bi]→ Rn (i = 1, 2), then length(γ1; [a1, b1]) = length(γ2; [a2, b2]).Thus we can define as length of Γ = γ([a, b]) the quantity

length(Γ) := length(γ; [a, b]) .

It is also well-known that, if Γ is a C1 regular curve, that is there exists a C1

parametrization γ : [a, b]→ Rn of Γ with |γ′(t)| 6= 0 for each t ∈ [a, b], then

(1.71) length(γ; [c, d]) =

∫ d

c

|γ′(t)| dt ∀[c, d] ⊂ [a, b] .

Let us recall that the 1-dimensional Hausdorff measure of a set E ⊂ Rn is definedas

H1(E) := limδ→0H1δ(E) = sup

δ∈(0,∞)

H1δ(E) ,

39

where

H1δ(E) = inf

∞∑i=1

diam(Ei) : E ⊂∞⋃i=1

Ei, diam(Ei) 6 δ

.

Given a curve Γ, it holds that

(1.72) H1(Γ) = length(Γ) ,

whether length(Γ) is finite or not (see Theorem 3.25).Assume now Γ = γ([a, b]) is C1 regular curve and define the measures

ν(E) := H1 Γ(E) := H1(Γ ∩ E) if E ∈ B(Rn) ,

µ(E) :=

∫E

|γ′(t)| dt if E ∈ B([a, b]) .

Then, by (1.71) and (1.72), it follows that ν and µ are finite Radon measure respec-tively on (Rn,B(Rn)) and ([a, b],B([a, b]). Moreover, according to Definition 1.95,

(1.73) ν = γ#µ .

In particular, by (1.73) and (1.69), it follows that

(1.74)

∫Γ

ϕdH1 =

∫Rnϕdν =

∫ b

a

ϕ γ dµ =

∫ b

a

ϕ(γ(t)) |γ′(t)| dt

for each ϕ ∈ C0c(Rn).

We consider now two measure spaces and see the resulting structure on their carte-sian product. In particular we introduce Fubini and Tonelli theorems which general-izes the notion of iterated integration of Riemannian calculus.

Definition 1.98 (Product σ-algebra). Let (X,M) and (Y,N ) be measure spaces.The product σ-algebra of M and N denoted byM×N is the σ-algebra generated inX × Y by

G = E × F : E ∈M, F ∈ N .

Remark 1.99. Let S ∈ M×N ; then for every x ∈ X the section Sx := y ∈ Y :(x, y) ∈ S belongs to N and for every y ∈ Y the section Sy := x ∈ X : (x, y) ∈ Sbelongs to M. In fact it is easily checked that the collections

SX := S ∈M×N : Sx ∈ N , ∀x ∈ X , SY := S ∈M×N : Sy ∈M, ∀y ∈ Y are σ-algebras in X×Y and contain G. Thus SX = SY = M×N (see [R1, Theorem8.2]).

Remark 1.100. If (X,M) and (Y,N ) are complete measure spaces (see Remark1.6), it need not be true that the product algebra M × N is complete. Indeed,suppose there exists E ∈M, E 6= ∅ with µ(E) = 0; and suppose there exists F ⊂ Ysuch that F /∈ N . Then E × F ⊂ E × Y , (µ× ν)(E × Y ) = 0, but, by Remark 1.99,E × F /∈ M×N . Therefore we will consider the completion (M×N )∗ in place ofM×N .

Theorem 1.101 (Fubini). Suppose that (X,M, µ) and (Y,N , ν) are complete σ-finitemeasure spaces.

40

(i) There exists a unique σ-finite measure on (X × Y, (M×N )∗), denoted withµ× ν and called product measure between µ and ν, such that

(µ× ν)(E × F ) = µ(E) ν(F ) ∀E ∈M, F ∈ N ,

(where we define 0∞ = ∞ 0 = 0).(ii) If S ∈ (M1 ×N )∗ then

Sy := x ∈ X : (x, y) ∈ X × Y ∈ N for ν-a.e. y ∈ Y ,

Sx := y ∈ Y : (x, y) ∈ X × Y ∈ N for µ-a.e. x ∈ X ,

y 7→ µ(Sy) is N -measurable, x 7→ µ(Sx) is M-measurable ,

(µ× ν)(S) =

∫X

ν(Sx) dµ(x) =

∫X

(∫Y

χS(x, y) dν(y)

)dµ(x)

=

∫Y

µ(Sy) dν(y) =

∫Y

(∫X

χS(x, y) dµ(x)

)dν(y) .

(iii) If u ∈ L1(X × Y, (M×N )∗, µ× ν) then

y 7→ u(x, y) is ν-integrable for µ-a.e. x ∈ X,

x 7→ u(x, y) is µ-integrable for ν-a.e. y ∈ Y,∫X×Y

u d(µ× ν) =

∫X

(∫Y

u(x, y) dν(y)

)dµ(x)

=

∫Y

(∫X

u(x, y) dµ(x)

)dν(y) .

Theorem 1.102 (Tonelli). Under the same assumptions of Theorem 1.101, if f :X × Y → [0,∞] is (M×N )∗-measurable, then

y 7→ u(x, y) is N -measurable for µ-a.e. x ∈ X ,

x 7→ u(x, y) is M-measurable for ν-a.e. y ∈ Y ,

x 7→∫Y

u(x, y) dν(y) is M-measurable,

y 7→∫X

u(x, y) dµ(x) is N -measurable,

and∫X×Y

u d(µ× ν) =

∫X

(∫Y

u(x, y) dν(y)

)dµ(x) =

∫Y

(∫X

u(x, y) dµ(x)

)dν(y)

in the sense that either both expressions are infinite or both are finite and equal.

For the proofs of Theorems 1.101 and 1.102 we refer to [R1, Theorems 8.8 and 8.12]or [GZ, Theorem 6.46 and Corollary 6.47].

41

Example 1.103 (Counterexample to Fubini-Tonelli’s theorem). Fubini and Tonelli’stheorems may fail when µ or ν is not σ-finite. Let X1 = X2 = [0, 1] ,M :=M1∩ [0, 1]and N := P([0, 1]) be respectively the class of Lebesgue measurable sets and of allsubsets in [0, 1], µ = L1 [0, 1] and ν = # be respectively the Lebesgue measure andthe counting measure in [0, 1]. Let u : [0, 1]2 → [0,∞), u(x, y) := χD(x, y), whereD := (x, y) ∈ [0, 1]2 : x = y . Then u is (M×N )-measurable, since D ∈ M×N .Indeed D = ∩∞h=1Qh with Qh := ∪hi=1[(i− 1)/h, i/h]2 ∈M×N . On the other hand∫

X

u(x, y) dµ(x) = 0 ∀y ∈ [0, 1],

∫Y

u(x, y) dν(y) = 1∀x ∈ [0, 1],

so that ∫Y

(∫X

u(x, y) dµ(x)

)dν(y) = 0,

∫X

(∫Y

u(x, y) dν(y)

)= 1 .

The failure of Theorems 1.101 (ii) and 1.102 is due to the fact that ν is not σ-additive.

An interesting application of Fubini and Tonelli’s theorems is the following Cava-lieri’s principle.

Proposition 1.104 (Cavalieri’s principle). Let (X,M) be a measure space, µ a pos-itive measure on it and u : X → [0,∞] be measurable. Let [0,∞) 3 t 7→ µ(u > t)denote the distribution function of u, that is,

(1.75) µ(u > t = µ (x ∈ X : u(x) > t) if t ∈ [0,∞) .

Let θ : [0,∞)→ [0,∞) be (strictly) increasing such that θ(0) = 0, θ : [0, T ]→ [0,∞)is absolutely continuous for each T ∈ [0,∞). Then

(1.76)

∫X

(θ u) dµ =

∫ ∞0

θ′(t)µ(u > t) dt .

In particular, if θ(t) = tp with p > 1, then∫X

up dµ = p

∫ ∞0

tp−1 µ(u > t) dt .

Proof. Let S := x ∈ X : u(x) > 0 and consider the restriction measure of µ to Sµ∗ := µ S.

1st step: Assume that µ∗ is not σ-finite. Then it is easy to see that both sidesare ∞. Indeed, since S := ∪∞h=1x ∈ X : u(x) > 1/h, there exists an integer h suchthat µ(u > 1/h =∞. Thus∫

X

θ u dµ >∫u> 1/h

θ u dµ > θ

(1

h

)µ(u > 1/h) =∞

= µ(u > 1/h∫ 1/h

0

θ′(t) dt 6∫ ∞

0

θ′(t)µ(u > t) dt .

2nd step: Assume that µ∗ is σ-finite. Then, since∫X

θ u dµ =

∫X

θ u dµ∗ and

∫ ∞0

θ′(t)µ(u > t dt =

∫ ∞0

θ′(t)µ∗(u > t) dt ,

we can assume that µ itself is σ-finite. Let

E := (x, t) ∈ X × [0,∞) : u(x) > t .

42

Et := x ∈ X : u(x) > t if t ∈ [0,∞) .

Then it can be proved that E ∈M×B([0,∞)) and the distribution function of u isthen

µ(Et) =

∫X

χE(x, t) dµ(x) ∀t ∈ [0,∞) .

Therefore, the right side of (1.76) is, by applying Fubini’s theorem with µ× L1,∫ ∞0

µ(Et) θ′(t) dt =

∫ ∞0

(∫X

χE(x, t) θ′(t) dµ(x)

)dt

=

∫X

(∫ ∞0

χE(x, t) θ′(t) dt

)dµ(x) .

(1.77)

For a given x ∈ X, observe that θ : [0, T ] → R is absolutely continuous for each0 6 T < u(x) 6 ∞. Thus it follows that∫ ∞

0

χE(x, t) θ′(t) dt =

∫ u(x)

0

θ′(t) dt = limT→u(x)−

∫ T

0

θ′(t) dt

= limT→u(x)−

θ(T ) = θ(u(x)) .(1.78)

From (1.77) and (1.78), (1.76) follows.

1.7. Weak*-convergence of measures. Regularization of Radon measuresin Rn.

Weak*-convergence of measures.

In this section we will assume that (X, d) is a l.c.s. metric space, the characteri-zation of the spaces of measures (Mloc(X))m and (M(X))m (see Corollary 1.78 andTheorem 1.83) as dual spaces induce on them a natural notion of weak*-convergence.

Definition 1.105. Let (X, d) be a l.c.s. metric space.

(i) Let (νh)h and ν be measures in (Mloc(X))m; we say that (νh)h locally weakly* converges

to ν, and write νh∗ ν, if

limh→∞

∫X

u dνh =

∫X

u dν ∀u ∈ C0c(X) .

(ii) Let (νh)h and ν be measures in (M(X))m; we say that (νh)h weakly* convergesto ν if

limh→∞

∫X

u dνh =

∫X

u dν ∀u ∈ C00(X) .

Remark 1.106. The weak*-convergence of finite Radon measures is called vagueconvergence, mainly in probability, and it is of considerable importance in applica-tions.

Remark 1.107. Weak* convergence of (νh)h ⊂M(X) to ν ∈M(X) does not implythat limh→∞ νh(A) = ν(A), even when A = X as the following exercise shows.

Exercise: Let X = R and let νh := δh. Then prove that (νh)h weakly* converges toν ≡ 0, but limh→∞ νh(R) = 1 6= 0 = ν(R).

43

Remark 1.108. The local weak* convergence is unique. More precisely

Exercise: Given (νh)h ⊂ (Mloc(X))m and νi ∈ (M(X))m i = 1, 2. If νh∗νi as

h→∞ for i = 1, 2, then ν1(E) = ν2(E) for each E ∈ Bcomp(X).

(Hint: If νh :=(ν

(1)h , . . . , ν

(m)h

), νi :=

(ν

(1)i , . . . , ν

(m)i

): Bcomp(X) → Rm, let define

the functionals Lh, Li : (C0c(X))m → R, if ψ = (ψ1, . . . , ψm),

Lh(ψ) :=

∫X

m∑j=1

ψj dν(j)h , Li(ψ) :=

∫X

m∑j=1

ψj dν(j)i .

Prove that:

• both Lh and Li are linear continuous functionals according to Definition 1.69;• L1(ψ) = L2(ψ) for all ψ ∈ (C0

c(X))m.

Applying Riesz representation theorem 1.73, conclude that ν1 = ν2. )

Proposition 1.109 (Locally weak* convergence vs. weak* convergence). Assumethat (νh)h, ν ⊂Mloc(X) . Then they are equivalent:

(i) νh∗ ν and suph |νh|(X) < ∞;

(ii) (νh)h, ν ⊂M(X) and (νh)h weakly* converges to ν.

.

Proof. (ii)⇒ (i): it is trivial, by definition, that νh∗ ν. By the characterization of

M(X) (see Theorem 1.83)

(1.79) |νh|(X) = ‖νh‖ = ‖Lνh‖(C00(X),‖·‖∞)′

and(1.80)(νh)h weakly* converges to ν ⇐⇒ (Lνh)h weakly* converges to Lν in (C0

0(X), ‖·‖∞)′ .

Thus, by (1.80),(Lνh)h is bounded in (C00(X), ‖ · ‖∞)′ and, by (1.79), it follows that

suph |νh|(X) < ∞.(i)⇒ (ii): By arguing as in the previuos implication, we have that the sequence

(Lνh)h is bounded in (C00(X), ‖ · ‖∞)′. By the sequential weak*-compactness of

bounded sets of (C00(X), ‖ · ‖∞)′ and Theorem 1.83, we have that, up to a subse-

quence,

(1.81) (Lνh)h weakly* converges to Lν∗ in (C00(X), ‖ · ‖∞)′ .

for some ν∗ ∈M(X). By the assumptions and (1.81), we can infer that

Lν(u) = Lν∗(u) ∀u ∈ C0c(X) .

Since C0c(X) is dense in (C0

0(X), ‖ · ‖∞), by the previous identity, we get that ν = ν∗

and then the desired conclusion.

Remark 1.110. The weak*-convergence of Radon measures if stable with respectto the push-forward of Radon measures. Indeed, let X, Y be l.c.s. metric spaces,f : X → Y continuous and proper: the continuity of f ensures that f−1(B) wheneverB ∈ B(Y ), and since f is proper the spaces C0

c(Y ) and C00(Y ) are continuously

mapped in C0c(X) and C0

0(X) respectively by u 7→ u f . Thus, if the sequence(νh)h of Radon measures on X locally weakly* converges to the measure ν, then the

44

sequence (f#νh)h locally weakly* converges to f#ν, and the same statement holds forfinite Radon measures and weak*-convergence.

Example 1.111 (Blow-ups of a curve in Rn). A fundamental idea in GMT is theexistence of tangent spaces to irregular submanifolds in terms of weak*-convergenceof suitable Radon measures. This idea will be developed later and we now sketch itwith an example. Let Γ be a C1 regular curve of Rn, that is Γ = γ([a, b]) for a C1

curve γ : [a, b] → Rn, injective with |γ′(t)| 6= 0 for each t ∈ [a, b]. Given t0 ∈ (a, b),the tangent space to Γ at x0 = γ(t0) is the line π = s γ′(t0) : s ∈ R. Consider nowΓ as a Radon measure, looking at ν = H1 Γ, and define the blow-ups νx0,r of ν atx0, setting

νx0,r :=1

r(Φx0,r)#(H1 Γ) ,

with Φx0,r(y) :=y − x0

rif y ∈ Rn.

Exercise: Prove that

νxo,r(E) = H1

(Γ− x0

r

)(E) ∀E ∈ B(Rn) .

(Hint. Use (1.69) and (1.73).)

Let us check that the property of π to be the tangent space implies that νx0,r∗

H1 π. Indeed, by (1.74), if ϕ ∈ C0c(R),∫

Rnϕdνx0,r =

1

r

∫Γ

ϕ

(y − x0

r

)dH1(y) =

1

r

∫ b

a

ϕ

(γ(t)− γ(t0)

r

)|γ′(t)| dt

=

∫ (b−t0)/r

−(t0−a)/r

ϕ

(γ(t0 + rs)− γ(t0)

r

)|γ′(t0 + rs)| ds

→∫Rϕ (s γ′(t0)) |γ′(t0)| ds =

∫π

ϕdH1 , as r → 0+ .

Other interesting examples of weak*-converging sequences of measures illustratinga wide variety of behaviours can be found in [Mag, Examples 4.20-4.23].

We now characterize the local weak*-convergence of positive Radon measures interms of evaluation on sets and introduce a criterion about the narrow convergenceof positive finite Radon measures.

Let us first introduce an useful criterion about foliations of Borel sets.

Lemma 1.112 (Foliations of Borel sets for positive Radon measures). Let (X, d) bea l.c.s. metric space. If Ett∈I is a disjoint family of Borel sets in X, indexed oversome set I, and µ is a positive Radon measure on (X,B(X)). Then µ(Et) > 0 for atmost countably many t ∈ I.

Proof. By Lemma 1.17, we can assume there exists an increasing sequence of compactsets (Kh)h of (X, d) such that X = ∪∞h=1Kh. Let Ih := t ∈ I : µ(Et ∩Kh) > 1/h.Observe that

(1.82) t ∈ I : µ(Et) > 0 = ∪∞h=1Ih .

45

Indeed, it is trivial that ∪∞h=1Ih ⊂ t ∈ I : µ(Et) > 0. Let us then prove the reverseinclusion. Assume that µ(Et) > 0 for some t ∈ I. Then there exists an integer h0

such that µ(Et) > 1/h0. On the other hand, since limh→∞ µ(Et ∩Kh) = µ(Et), thereexists an integer h > h0 such that µ(Et∩Kh) > 1/h0 > 1/h. Thus t ∈ Ih and (1.82)follows.

Let us now show that Ih is finite with #(Ih) 6 hµ(Kh). Therefore, by (1.82), theproof will be accomplished. Let J ⊂ Ih be finite. Then

µ(Kh) > µ (∪t∈Ih(Et ∩Kh)) > µ (∪t∈J(Et ∩Kh)) =∑t∈J

µ((Et ∩Kh)) >#(J)

h.

Theorem 1.113 (Characterization of the locally weak* convergence of positive Radonmeasures). Let (µh)h and µ be positive Radon measures on (X,B(X)).Then the fol-lowing are equivalent.

(i) µh∗ µ as h→∞.

(ii) If K compact and A open, then

(1.83) µ(K) > lim suph→∞

µh(K) ,

(1.84) µ(A) 6 lim infh→∞

µh(A) .

(iii) If E ∈ Bcomp(X) with µ(∂E) = 0, then

µ(E) = limh→∞

µh(E) .

Moreover, if µh∗µ as →∞, then for every x ∈ sptµ there exists (xh)h ⊂ X with

(1.85) limh→∞

xh = x, xh ∈ sptµh ∀h ∈ N .

Proof. (i)⇒ (ii): Let K ′ and A′ be respectively a compact and open set such that

K ′ ⊂ A′. By Urysohn’s lemma 1.65 there exists a function ϕ ∈ C0c(X) such that

χK′ 6 ϕ 6 χA′ , then

µh(K′) 6

∫X

ϕdµh 6 µh(A′)∀h, µ(K ′) 6

∫X

ϕdµ 6 µ(A′) .

By (i) and the previous inequality we have

(1.86) lim suph→∞

µh(K′) 6

∫X

ϕdµ 6 µ(A′) ,

(1.87) lim infh→∞

µh(A′) >

∫X

ϕdµ > µ(K ′) .

Setting K ′ = K in (1.86) and passing to the infimum on all open sets A′ ⊃ K,by Theorem 1.14 (i), (1.83) follows. Setting A′ = A in (1.87) and passing to thesupremum on all compact sets K ′ ⊂ A, by Theorem 1.14 (ii), (1.84) follows.

(ii)⇒ (iii): Notice that

µ(E) 6 µ(E) 6 µ(E) = µ(

E) + µ(∂E) = µ(

E) ,

46

then, since E is compact, combining (ii) and the monotonicity of µ

µ(E) = µ(E) 6 lim inf

h→∞µh(

E) 6 lim inf

h→∞µh(E)

6 lim suph→∞

µh(E) 6 lim suph→∞

µh(E) 6 µ(E) = µ(E) .

In particular it follows that µ(E) = lim infh→∞ µh(E) = lim suph→∞ µh(E), and thenthe desired conclusion.

(iii)⇒ (i): Let ϕ ∈ C0c(X), ϕ > 0. By Lemma 1.112, there exists I ⊂ [0,∞) such

that L1([0,∞) \ I) = 0µ(ϕ = t) = 0 ∀t ∈ I .

Being ϕ continuous,∂ϕ > t ⊆ ϕ = t ∀t ∈ [0,∞) .

Hence (iii) implies that

µ(ϕ > t) = limh→∞

µh(ϕ > t) ∀t ∈ I .

Let fh, f : [0,∞)→ R (h = 1, 2, . . . ), fh(t) := µh(ϕ > t), and f(t) := µ(ϕ > t).Then, being fh and f nondecreasing functions, fh and f are Borel measurable and

limh→∞

fh(t) = f(t) L1-a.e. t ∈ [0,∞] ,

|fh(t)| 6 µ(sptϕ)χ[0,supRn ϕ](t) ∀t ∈ [0,∞) .

By dominated convergence theorem and Cavalieri’s principle (1.76) with θ(t) = t, wehave ∫

X

ϕdµ =

∫ ∞0

µ(ϕ > t) dt = limh→∞

∫ ∞0

µh(ϕ > t) dt = limh→∞

∫X

ϕdµh .

If ϕ has a general sign, we can decompose it as ϕ = ϕ+−ϕ− and apply the previousargument to ϕ+ and ϕ−.

We finally prove (1.85) and let us prove that for every ε > 0 there exists h ∈ Nsuch that sptµh∩B(x, ε) 6= ∅. By contradiction, there exists ε0 > 0 and an increasingsequence of integers (hk)h with limk→∞ hk = ∞ such that sptµhk ∩ B(x, ε0) = ∅ foreach k. By (1.84), it follows that

µ(B(x, ε0)) 6 lim infk→∞

µhk(B(x, ε0)) = 0 ,

which contradicts the fact that x ∈ sptµ and then µ(B(x, ε0)) > 0.

Remark 1.114. (Limits points of support points and uniform lower bounds [Mag,

Remark 4.28]) If µh∗µ, xh ∈ sptµh for every h ∈ N, and xh → x, then it is not true,

in general, that x ∈ sptµ. For instance, if X = R, the sequences

µh =

(1− 1

h

)δ1 +

1

hδ1/h, xh =

1

h.

The implication becomes true as soon as some kind of uniform lower bound on themeasure assigned by the µh around their support points is assumed. More precisely,let (µh)h be a sequence of positive Radon measures on X, such that, for every r > 0,

(1.88) lim suph→∞

inf µh(B(x, r)) : x ∈ sptµh > 0 .

47

Under this assumption, we claim that, if µh∗µ, xh → x, and xh ∈ sptµh , then

x ∈ sptµ. Indeed, let c(r) denote the left-hand side of (1.88). For every r > 0, leth0 ∈ N be such that B(xh, r) ⊂ B(x, 2r) for every h > h0. By (1.83), and if necessaryextracting a subsequence so as to exploit (1.88),

µ(B(x, 2r)) > lim suph→∞

µh(B(x, 2r)) > lim suph→∞

µh(B(xh, r)) > c(r) > 0 .

By the arbitrariness of r, we find that x ∈ sptµ.

Proposition 1.115 (Characterization of the narrow convergence of positive Radonmeasures). Let (µh)h be a sequence of positive, finite Radon measures on (X,B(X))and assume the existence of a positive, finite Radon measure µ such that

(1.89) limh→∞

µh(X) = µ(X) and lim infh→∞

µh(A) > µ(A)

for every A ⊂ X open set. Then

(NC) limh→∞

∫X

u dµh =

∫X

u dµ ∀u ∈ C0b(X)

where C0b(X) denotes the class of all bounded continuous function u : X → R. In

particular (µh)h weakly* converges to µ. Moreover if (NC) holds so does (1.89), thatis (NC) and (1.89) are equivalent.

Proof. Let u ∈ C0b(X). Possibly replacing u by αu+ β for suitable α, β ∈ R, we can

assume without loss of generality that 0 6 u 6 1. We first show that

(1.90) lim infh→∞

∫X

v dµh >∫X

v dµ ,

for each continuous function v : X → [0,∞). Indeed, by Cavalieri’s principle (1.76)with θ(t) = t and Fatou lemma, we infer

lim infh→∞

∫X

v dµh = lim infh→∞

∫ ∞0

µh(v > t) dt >∫ ∞

0

lim infh→∞

µh(v > t) dt

>∫ ∞

0

µ(v > t) dt .

Let us recall the followingExercise: Prove that, if (ah) and (bh)h are sequences of real numbers such that

a 6 lim infh→∞

ah, b 6 lim infh→∞

bh, lim suph→∞

(ah + bh) 6 a+ b ,

for some a, b ∈ R, then there exist limh→∞ ah = a and limh→∞ bh = b.Setting

ah =

∫X

u dµh, a =

∫X

u dµ, bh =

∫X

(1− u) dµh, b =

∫X

(1− u) dµ ,

from (1.90) and the assumption limh→∞ µh(X) = µ(X), the proof is accomplished.Finally, if (NC) holds, by choosing as test function ϕ ≡ 1 in (NC), we get that

µ(X) = limh→∞

µh(X) .

48

On the other hand, since (NC) also implies the local weak* convergence of (µh)h toµ, by Theorem 1.113, it follows that, for each open set A ⊂ X,

lim infh→∞

µh(A) > µ(A) .

Thus (1.89) follows.

Remark 1.116. Convergence (NC) is called narrow convergence of (µh)h to µ, andsometimes (by abuse of notation) weak-convergence of (µh)h to µ. It is a stronger con-vergence than the weak*-convergence: indeed, for instance, limh→∞ µh(X) = µ(X) isnot granted for the weak*-convergence (recall the previous exercise). Note also that

C0c(X) ⊂ C0

0(X) ⊂ C0b(X) ,

and C0c(X) = C0

0(X) = C0b(X) = C0(X) if X is compact, and in this case the two

notions of convergence coincide. On the other hand, for non-compact X the spaceM(X) is not the dual of (C0

b(X), ‖ · ‖∞) as the following exercise shows.Exercise: Let X = R and let F := f ∈ C0(R) : ∃ f(∞) := lim|x|→∞ f(x) ∈ R.

(i) Prove that F is a closed subspace of (C0b(R), ‖ ·‖∞) and C0

c(R) ⊂ C00(R) ⊂ F .

(ii) Let L : F → R be the linear functional defined as L(f) := f(∞) and provethat it can be extended to a functional L ∈ (C0

b(R), ‖ · ‖∞)′ by means of theHahn-Banach theorem.

(iii) Prove that there is no a finite Radon measure ν on R, that is an elementν ∈ M(R), such that L(f) =

∫R f dν for each f ∈ C0

b(R). (Hint: Observe

that L(f) = 0 for each f ∈ C0c(R)).

We now consider the local weak*-convergence of Radon vector measures and wepoint out some relationships with the local weak*-convergence of their total variation.Before let us introduce an alternative approximation to Theorem 1.14 for positiveRadon measures, which will need in the proof.

Lemma 1.117. Let (X, d) be a l.c.s. metric space, let µ be a positive Radon measureon (X,B(X)) and let E ∈ Bcomp(X) with µ(∂E) = 0. Then, for each ε > 0 thereexist a compact set K and an open set A (which may be empty) such that

A ⊂ E ⊂K and µ(K \ A) < ε .

Remark 1.118. Lemma 1.117 is a refinement of the approximation of Radon mea-sures on a l.c.s metric space by means of compact sets from below and open sets fromabove (see Theorem 1.14).

Proof. Let us first observe that, since E is compact, by Lemma 1.17, there is arelatively compact open set V such that

(1.91) E ⊂ V ⊂ V .

If h, m are integers andE 6= ∅ (otherwise choose A = ∅) , let us consider the

sequences of sets

Ah :=

x ∈

E : d(x, ∂E) >

1

h

, Km :=

x ∈ V : d(x, E) 6

1

m

.

49

Since (Ah)h is an increasing sequence of open sets, with Ah ⊂E and

E = ∪∞h=1Ah, it

follows that

limh→∞

µ(Ah) = µ(E) = µ(E) <∞ .

Thus, if we take A = Ah for h large enough, then, µ(E \A) < ε/2. In the same way,

(1.92) (Km)m is a decreasing sequence of compact sets, ∩∞m=1Km = E ,

with

(1.93)Km ⊃ E .

Indeed it is immediate, by construction, that (Km)m is a decreasing sequence of closedsets and, by (1.91), that ∩∞m=1Km = E. Since each Km ⊂ V and V is compact, Km

is compact, too and then (1.92). On the other hand, if x0 ∈ E, then, by (1.91),x0 ∈ V and d(x0, E) = 0 < 1/m for each m. Since V is open and the function

V 3 x 7→ d(x, E) is continuous, it follows that x0 ∈Km for each m and then (1.93).

Since µ(K1) <∞, by (1.92),

limm→∞

µ(Km) = µ(E) = µ(E) ,

and, if we choose K = Km for m large enough, µ(K \ E) < ε/2 withK ⊃ E, by

(1.93).

Theorem 1.119. Let (νh)h and ν be Rm-valued Radon vector measures, that is νh, ν :B(X)→ Rm, and let µ a positive Radon measure on a l.c.s. metric space (X, d).

(i) If νh∗ν, then for every open set A ⊂ X

(1.94) |ν|(A) 6 lim infh→∞

|νh|(A) .

(ii) If νh∗ν and |νh|

∗µ, then

(1.95) |ν|(B) 6 µ(B) ∀E ∈ B(X) .

Moreover, if E ∈ Bcomp(X) with µ(∂E) = 0, then

ν(E) = limh→∞

νh(E) .

(iii) If νh∗ν and limh→∞ |νh|(X) = |ν|(X) < ∞, then (NC) holds with µh = |νh|

and µ = |ν|. In particular |νh|∗|ν|.

Proof. (i): Let us recall that, by Riesz representation theorem 1.73, νh = (ν1h, . . . , ν

mh ) =

wνh |νh|, ν = (ν1, . . . , νm) = wν |ν| with wνh , wν : X → Rm Borel measurable,|wνh| = |wν | = 1, |νh|-a.e. and |ν|-a.e. in X, respectively, and, for each open setA ⊂ X,

(1.96) |ν|(A) = sup

∫X

(wu, ψ)Rm d|ν| : ψ ∈ (C0c(A))m, ‖ψ‖∞ 6 1

.

By the assumptions, for each ϕ ∈ C0c(X)∫

X

ϕwν d|ν| =∫X

ϕdν = limh→∞

∫X

ϕdνh = limh→∞

∫X

ϕwνh d|νh| ∀ϕ ∈ C0c(X) .

50

This implies that for each ψ ∈ (C0c(A))m with ‖ψ‖∞ 6 1∫

X

(wν , ψ)Rm d|ν| = limh→∞

∫X

(wνh , ψ)Rm d|νh| 6 lim infh→∞

|νh|(A)

Since ψ ∈ (C0c(A))m with ‖ψ‖∞ 6 1 is arbitrary, by (1.96), (1.94) follows.

(ii): Let A ⊂ X be a relatively compact open set and, At := x ∈ A : d(x, ∂A) > tand let u ∈ C0

c(A) such that χAt 6 u 6 χA. Then, by (1.94), we have

|ν|(At) 6 lim infh→∞

|νh|(At) 6 lim infh→∞

∫X

u d|νh| =∫X

u dµ 6 µ(A) .

By letting t → 0+, we get |ν|(A) 6 µ(A), and since A is arbitrary, inequality (1.95)follows from Theorem 1.14. Let us now prove that ν(E) = limh→∞ νh(E) wheneverE ∈ Bcomp(X) with µ(∂B) = 0. Given ε > 0, by Lemma 1.117, we find an open set

A and a compact set K such that A ⊂ E ⊂K and µ(K \ A) 6 ε. Then, for every

u ∈ C0c(K), 0 6 u 6 1 with u = 1 on A, we find∣∣∣∣∫

X

u dνh − νh(E)

∣∣∣∣ 6 ∫X

|u− χE| d|νh| 6 |νh|(K \ A) ,∣∣∣∣∫X

u dν − ν(E)

∣∣∣∣ 6 |ν|(K \ A) 6 µ(K \ A) ,

limh→∞

∣∣∣∣∫X

u dνh −∫X

u dν

∣∣∣∣ = 0 .

Since |νh|∗µ and K \ A is compact, by (1.83), we have lim suph→∞ |νh|(K \ A) 6

|ν|(K \ A) 6 µ(K \ A). Recalling µ(K \ A) 6 ε, we thus conclude that

lim suph→∞

|νh(E)− ν(E)| 6 lim suph→∞

∣∣∣∣νh(E)−∫X

u dνh

∣∣∣∣+ lim suph→∞

∣∣∣∣∫X

u dνh −∫X

u dν

∣∣∣∣+ lim sup

h→∞

∣∣∣∣∫X

u dν − ν(E)

∣∣∣∣ 6 2ε, ∀ ε > 0 .

(iii): Without loss of generality, we can assume that both |νh|(X) < ∞ for eachh and |ν|(X) < ∞, that is that (νh)h and ν are measures contained in (M(X))m.Thus, by claim (i), it follows that the assumptions of Proposition 1.115 are satisfiedand then the proof is accomplished.

Remark 1.120. A typical application of statement (ii) of Theorem 1.119 (or also ofstatement (iii) of Theorem 1.113) is the following: let us consider an increasing family(At)t of relatively compact open sets labelled on an interval I such that As ⊂ At, fors < t. Then,

Exercise: µ(∂At) = 0 except for countably many t ∈ I.(Hint: Let (Vi)i be an increasing sequence of relaatively compact open sets such thatX = ∪∞i=1Vi (see Lemma 1.17). Since ∂At with t ∈ I are pairwise disjoint, by theadditivity of µ, prove that , for given ε > 0 and i ∈ N the set

t ∈ I : µ(∂At) > ε, At ⊂ Vi .Then deduce the desired conclusion.)

51

Hence, by Theorem 1.119 (ii) νh(At)→ ν(At) for L1-a.e. t ∈ I.

The classical De La Vallee Poussin compactness criterion for finite Radon measureseasily follows by the sequential weak*-compactness of bounded sets in a dual spaceof a separable normed vector space (see, for instance, [SC, Theorem 3.30]) and thecharacterization of the space of finite Radon measures (Theorem 1.83).

Theorem 1.121 (Weak*-compactness). If (νh)h is a sequence of Rm- valued fi-nite Radon measures on the l.c.s. metric space X, that is (νh)h ⊂ (M(X))m, withsuph |νh|(X) < ∞, then it has a weakly*- converging subsequence. Moreover, themap ν 7→ |ν| is lower semicontinuous with respect to the weak*-convergence, that is,assume that (νh)h weakly*-converges to ν, then

|ν|(X) 6 lim infh→∞

|νh|(X) .

.

The previous theorem can be used to get immediately a corresponding result in theframe of local weak*-convergence.

Corollary 1.122 (Local weak* compactness). Let (νh)h be a sequence of Rm- valuedRadon measures on the l.c.s. metric space X, (νh)h ⊂ (Mloc(X))m, such that

sup|νh|(K) : h ∈ N <∞for every compact K ⊂ X; then it has a locally weakly*-converging subsequence.

Proof. Let (Vi)i be a sequence of relatively compact open sets such that

X = ∪∞i=1Vi and Vi ⊂ Vi+1 for each i ∈ N .For given i, let (νih)h be the sequence of Rm-valued finite Radon measures defined as

νih := νh Vi .

Since, by our assumption,

suph|νih|(X) = sup

h|νh|(Vi) < ∞ .

By Theorem 1.121 and by means of a standard diagonal process, there exist a subse-quence (hk)k and finite Radon measure νi ∈ (M(X))m such that (νhk Vi)k weakly*converges to νi, that is

(1.97)

∫Vi

ϕdνhk =

∫X

ϕd(νhk Vi)→∫X

ϕdνi for each ϕ ∈ C00(X), i ∈ N .

Moreover, by definition of Rm-valued finite Radon measure (see Definition 1.74 (ii)),thete exists a Borel measurable functions wi : X → Sm−1 such that

νi = wi|νi| with |νi| positive finite Radon measure on X, for each i ∈ N .Let us now prove that, for each i ∈ N,

νi Vi = νi+1 Vi ,

that is, there exists a finte positive Radon measure on X such that

(1.98) |νi| Vi = |νi+1| Vi = µ and wi(x) = wi+1(x) µ-a.e. x ∈ Vi .

52

Fix i ∈ N and let us consider a function ϕ ∈ C0c(Vi) . Then we can infer that∫

X

ϕdνhk Vi =

∫Vi

ϕdνhk =

∫Vi+1

ϕdνhk =

∫X

ϕd(νhk Vi+1) .

Passing to the limit as k →∞ in the previous identity, by (1.97), we get∫X

ϕwi d|νi| =

∫X

ϕdνi =

∫X

ϕdνi+1

=

∫X

ϕwi+1 d|νi+1| for each ϕ ∈ C0c(Vi), i ∈ N .

Let wi = (wi,1, . . . , wi,m), then we can rewrite the previous identity as

(1.99)

∫Vi

ϕwi,j d|νi| =∫X

ϕwi,j d|νi| =∫X

ϕwi+1,j d|νi+1| =∫Vi

ϕwi+1,j d|νi+1|

for each ϕ ∈ C0c(Vi), i ∈ N, j = 1, . . . ,m, which implies that, for each u = (u1, . . . , um) ∈

(C0c(Vi))

m and for each i ∈ N,

(1.100)

∫X

(wi, u)Rm d|νi| =∫X

(wi+1, u)Rm d|νi+1|

By (1.100) and the representation of the total variation of a Rm-valued Radon measure(see (1.33))we get that

(1.101) |νi|(A) = |νi+1|(A) for each open set A ⊂ Vi .

By the approxomiation with open sets for a positive Radon measure, we can inferthat

(1.102) |νi| Vi = |νi+1| Vi = µ .

Therefore, by (1.102), we can rewrite (1.99) as follows

(1.103)

∫Vi

ϕwi,j dµ =

∫Vi

ϕwi+1,j dµ for each ϕ ∈ C0c(Vi), i ∈ N, j = 1, . . . ,m .

By Remark 1.64 , for given i ∈ N, for each Borel set E ⊂ Vi there exists a sequence(ϕh)h ⊂ C0

c(Vi), such that

(1.104) ϕh → χE in L1(Vi, µ) and |ϕh(x)| 6 1 for each x ∈ Vi .By (1.103), (1.104) and the Lebesgue dominated convergence theorem, we can inferthat ∫

E

wi,j dµ =

∫E

wi+1,j dµ for each Borel set E ⊂ Vi, i ∈ N, j = 1, . . . ,m , .

which is equivalent to

(1.105) wi(x) = wi+1(x) µ-a.e. x ∈ Vi .By (1.102) and (1.105), (1.98) follows. Let us now define a positive Radon measureµ : B(X)→ [0,∞] and a Borel measurable function w : X → Sm−1 as follows:

µ(E) := |νi|(E) if E ∈ B(Vi) for some i

andw(x) := wi(x) if x ∈ Vi for some i .

53

By (1.98) both µ and w are well defined. Let us define the Rm-valued Radon measureν : Bcomp(X)→ Rm

ν(E) =

∫E

w dµ if E ∈ Bcomp(X) .

Then by (1.97), it follows that (νhk)k locally weak* converges to ν and we accomplishthe proof.

Regularization of Radon measures in Rn.

In this section we assume that X = Rn, endowed with the Euclidean metric, and weare going to deal with the approximation of a given Rm-valued Radon vector measureν on Rn by means of a sequence of Rm-valued Radon vector measures

νh = Lnwh on Rn

(see (1.19)) with wh ∈ C∞(Rn,Rm) and νh∗ν.

We saw that, given f ∈ Lp(Rn) with 1 6 p < ∞, there exists (fh)h ⊂ C0c(Rn)

such that fh → f in Lp(Rn) (see Theorem 1.63). Since the differential structure ofRn, we are going to improve this approximation, looking for an approximation byregular C∞-functions on Rn. A powerful tool for getting such a goal is the so-calledapproximation by convolution, which we will briefly recall here.

Let us first recall the notion of support for a (Lebesgue) measurable function.

Let us recall (see (1.6)) that given a function f : Rn → R, its support is the set

(S) spt(f) := x ∈ Ω : f(x) 6= 0 .This definition is not suitable for a (Lebesgue) measurable function f : Rn → R.Indeed we would like that this notion satisfies the following property:

f1 = f2 a.e. in Rn ⇒ spt(f1) = spt(f2) , except for a negligible set.

But this is not the case. Indeed

Example: Let f1 := χQ : R→ R and f2 ≡ 0. Then it is clear that

f1 = f2 a.e. in Rbut

spt(f1) = Q = R and spt(f2) = ∅ .

Proposition 1.123 (Essential support of a function). Let f : Rn → R. Denote

Af := ω ⊂ Rn : ω open set and f = 0 a.e. in ωand let

Af := ∪ω∈Afω .Then Af is an open set and

f = 0 a.e. inAf .

The closed set

(ES) spte(f) := Rn \ Afis called the essential support of f in Rn.

54

Proof of Proposition 1.123. See [B, Proposition 4.17]

Remark 1.124. (i) From definition (ES), it follows that, if f1 = f2 a.e. in Rn,then spte(f1) = spte(f2).

(ii) Definitions (S) and (ES) agree when the function is continuous. More preciselyExercise: If f : Rn → R is continuous, then

Rn \ Af = x ∈ Rn : f(x) 6= 0 .

Definition 1.125 (Friedrichs’ mollifiers, 1944). A sequence of mollifiers is a sequenceof functions %h : Rn → R (h = 1, 2, . . . ) such that, for each h,

(Mo1) %h ∈ C∞(Rn) ;

(Mo2) spt(%h) ⊂ B(1/h) ;

(Mo3)

∫Rn%h dx = 1 ;

(Mo4) %h(x) > 0 ∀x ∈ Rn .

Example of mollifiers: It is quite simple constructing a sequence of mollifiers,starting from a given non vanishing function % : Rn → R satisfying

% ∈ C∞c (Rn) , spt(%) ⊂ B(1), % > 0 .

For instance, let

%(x) :=

exp

(1

|x|2 − 1

)if |x| < 1

0 if |x| > 1.

Then it is easy to see that % ∈ C∞c (Rn). Moreover we yield a sequence of mollifiersby defining

(1.106) %h(x) := c hn %(hx) x ∈ Rn, h ∈ N

and

c :=

(∫Rn% dx

)−1

.

Remark 1.126. Observe that, without loss of generality, by (1.106), we can assumethat a sequence of mollifiers (%h)h satisfies the simmetry condition

(1.107) %h(−x) = %h(x) ∀x ∈ Rn, h ∈ N

Notation: If A, B ⊂ Rn, A±B denotes the set

A±B := a± b : a ∈ A, b ∈ B Exercise: Prove that(i) if A is compact and B is closed, then A+B is closed;(ii) if A and B are compact so is A+B

55

Proposition 1.127 (Definition and first mollifiers’ properties). Let f ∈ L1loc(Rn) and

let (%h)h be a sequence of mollifiers. Define, for given h ∈ N and x ∈ Rn,

fh(x) = (%h ∗ f)(x) :=

∫Rn%h(x− y) f(y) dy x ∈ Rn.

Then

(i) the function fh : Rn → R is well defined;(ii) fh(x) = (%h ∗ f)(x) = (f ∗ %h)(x) for all x ∈ Rn and h ∈ N;(iii) fh ∈ C0(Rn) for each h;(iv) if f ∈ C0(Rn), then %h ∗ f → f uniformly on compact sets of Rn, as h→∞.

The function fh is called hth- mollifier of f .

Proof. See [B, Propositions 4.19 and 4,21] and [SC, Proposition 2.68 and Lemma2.74]

Remark 1.128. The symbol ∗ denotes the convolution product between two func-tions defined on the whole Rn. Notice also that the conclusions of Proposition 1.127still holds if f ∈ L1

loc(Rn) and % ≡ %h ∈ C0(Rn) satisfying (Mo2). Actually, it ispossible to define the convolution product between two functions g ∈ Lp(Rn) with1 6 p 6 ∞ and f ∈ L1(Rn)

(g ∗ f)(x) :=

∫Rng(x− y) f(y) dy

and it holds that (see [GZ, Theorem 6.51])

(g ∗ f) ∈ Lp(Rn) and ‖g ∗ f‖Lp(Rn) 6 ‖g‖Lp(Rn) ‖f‖L1(Rn) .

Theorem 1.129 (Friedrichs-Sobolev, approximation by convolution in Lp). Let f ∈L1loc(Rn) and (%h)h be a sequence of mollifiers. Then

(i) f ∗ %h ∈ C∞(Rn) for each h ∈ N.(ii) ‖f ∗ %h‖Lp(Rn) 6 ‖f‖Lp(Rn) for each h ∈ N, f ∈ Lp(Rn), for every p ∈ [1,∞].(iii) spt(f ∗ %h) ⊂ spte(f) +B(1/h) for each h ∈ N.(iv) If f ∈ Lp(Rn) with 1 6 p 6 ∞, then f ∗ %h ∈ C∞(Rn) ∩ Lp(Rn) for each

h ∈ N, and f ∗ %h → f as h→∞, in Lp(Rn), provided that 1 6 p < ∞.

Proof of Theorem 1.129. See [B, Proposition 4.20 and Theorem 4.22] and [SC, The-orem 2.70].

Historical notes: Mollifiers were introduced by K. Friedrichs in 1944, which are,according to P. Lax, a watershed in the modern theory of PDEs. However, S. Sobolevhad used mollifiers in his epoch making 1938 paper [So] (the paper containing theproof of the Sobolev embedding theorem), as Friedrichs himself acknowledged in laterpapers.

Let us now come back to the approximation of a Radon vector measure. Let ν bea Rm-valued Radon vector measure and (%h)h be a sequence of mollifiers. Let us thendefine the sequence of functions ν ∗ %h : Rn → Rm (h ∈ N) as

(1.108) (ν ∗ %h)(x) :=

∫Rn%h(x− y) dν(y) if x ∈ Rn .

56

Theorem 1.130 (Approximation of Radon vector measures). Let ν = (ν1, . . . , νm)be a Radon vector measure in Rn and let (%h)h be a sequence of mollifiers which alsosatisfies simmetry condition (1.106). Then

(i) The functions ν ∗ %h : Rn → Rm belong to (C∞(Rn))m and ∇α(ν ∗ %h) =ν ∗ ∇α%h for any α ∈ Nn.

(ii) If νh := Lnν∗%h, the sequence of measures (νh)h locally weakly*converges in Rn

to ν as h→∞ and, for each E ∈ B(Rn), it holds the estimate

|νh|(E) =

∫E

|ν ∗ %h| dx 6 |ν|(I1/h(E)) .

(iii) The sequence of measures (|νh|)h locally weakly*converges in Rn to |ν| as h→∞.

Proof. (i): The first statement can be easily proved as in the proof of Theorem 1.129(i), by induction on the length of α by using a difference quotient argument andpassing to the limit under the integral.

(ii): Let us first note that, using Fubini’s theorem and simmetry condition (1.106),it is easily seen that

(1.109)

∫Rn

(ν ∗ %h) v dx =

∫Rn

(v ∗ %h) dν

for each v ∈ L1(Rn). Thus, if u ∈ C0c(Rn), by (1.109) and Proposition 1.127 (iv),∫

Rnu dνh =

∫Rnu (ν ∗ %h) dx =

∫Rn

(u ∗ %h) dν →∫Rnu dν as h→∞ .

Let E ∈ B(Rn) and let us estimate |νh|: by (1.20) and for Fubini’s theorem

|νh|(E) =

∫E

|ν ∗ %h| dx =

∫E

∣∣∣∣∫Rn%h(x− y) dν(y)

∣∣∣∣6∫E

(∫Rn%h(x− y) d|ν|(y)

)dx =

∫Rn

(∫E

%h(x− y) dx

)d|ν|(y)

6∫I1/h(E)

(∫E

%h(x− y) dx

)d|ν|(y) 6 |ν|(I1/h(E)) .

(iii): Let At := U(t) if t ∈ I := (0,∞). By Remark 1.120, we can find an increasingsequence of open sets Ak b Rn such that Rn = ∪∞k=1Ak and |ν|(∂Ak) = 0 for eachk ∈ N. As a consequence of (ii),

lim suph→∞

|νh|(Ak) 6 |ν|(I0(Ak)) = |ν|(Ak) = |ν|(Ak) for each k .

On the other hand, Theorem 1.119 (i) implies that lim infh→∞ |νh|(A) > |ν|(A) forany open set A ⊂ Rn. By Proposition 1.115, we infer that (|νh|)h weakly* convergesto |ν| in Ak, and since k is arbitrary the statement follows.

57

2. Differentiation of Radon measures ([AFP, Ma])

Motivation: In this section we are going to introduce the main results about thedifferentiation of measures which will be used later in the rectifiability and in thestudy of sets of finite perimeter. One of the main goals is to prove the followingresult.

Theorem 2.1 (of Lebesgue points). : Let µ be a positive Radon measure on (Rn,B(Rn))and let f ∈ L1

loc(Rn, µ). Then µ-a.e. x ∈ spt(µ) there exists

(LP) limr→0

1

µ(B(x, r))

∫B(x,r)

|f(y)− f(x)| dµ(y) = 0 .

Definition 2.2. A point x ∈ Rn for which (LP) holds is called a Lebesgue point off .

2.1. Covering theorems and Vitali-type covering property for measures onRn. We begin to introduce two types of covering theorems in X = Rn. The differencebetween them is that the first ones (Vitali’s coverings) apply to a larger class ofcoverings and a narrower class of measures whereas in the second type the coverings(Besicovitch’s coverings) are more restricted but the measures can be very general;for example all Radon measures on Rn are included. In both cases we first prove ageometric result on collections of balls in Rn and then apply it to get a Vitali-typecovering theorem for measures.

Let us begin with some notions on coverings in a general metric space.To begin with, let us agree that by disjoint family of subsets of a metric space

(X, d) we mean a family F such that E ∩ F = ∅ whenever E, F ∈ F and E 6= F ;we set also

∪F := ∪E∈FE .Definition 2.3. Let (X, d) be a metric space.

(i) A family F of closed balls of (X, d) is a cover of a set A ⊂ X if

A ⊆ ∪F .(ii) A family F of closed balls of (X, d) is a fine cover of A, or also that F covers

A in the sense of Vitali, if it is a cover of A and, for each x ∈ A,

(2.1) inf d(B) : B ∈ F , x ∈ B = 0 .

Vitali covering theorem and the Lebesgue measure.

Theorem 2.4 (Vitali covering theorem). Let G be a family of closed balls in Rn with

D = sup d(B) : B ∈ G < ∞ .

Then there exists a (pairwise) disjoint family F ⊆ G, which is at most countable,such that

∪B∈GB ⊂ ∪B∈FB .

where B is an enlargement of B, that is B = 5B.

Before the proof of Vitali’s covering theorem, let us recall the Hausdorff MaximalPrinciple (see [GZ, Theorem 1.4]).

58

Theorem 2.5 (Hausdorff Maximal Principle). If S is a family of sets (or a collectionof families of sets) and if ∪E : E ∈ E ∈ S for any subfamily E of S totally orderedwith respect to the inclusion, that is with the property that

E1 ⊂ E2 or E2 ⊂ E1 whenever E1, E2 ∈ E .Then there exists E∗ ∈ S, which is maximal in the sense that it is not a subset of anyother member of S.

Proof of Theorem 2.4. Let us define the sequence of subfamilies of G

Gj :=

B ∈ G :

D

2j< d(B) 6

D

2j−1

j = 1, 2, . . .

Then it is trivial to see that

(2.2) G = ∪∞j=1Gj ,

(2.3) Gj ∩ Gj′ = ∅ ∀j 6= j′ .

Let us define inductively a subfamily Fj ⊂ Gj as follows.Let F1 ⊂ G1 be a maximal subcolletion of pairwise disjoint elements if G1 6= ∅. For

if not, let F1 = ∅.Exercise: Prove that such a family F1 exists by means of the Hausdorff Maximal

Principle.Assuming that F1,F2, . . . ,Fj−1 have been chosen, let Fj be a maximal pairwise

disjoint family of the family

G∗j :=B ∈ Gj : B ∩B′ = ∅, ∀B′ ∈ ∪j−1

i=1Fi

if G∗j 6= ∅; for if not let Fj := ∅. Let

F = ∪∞j=1Fj .From (2.2) and (2.3), F ⊂ G and F is a disjoint collection of balls. Moreover F is at

most countable, since the pairwise disjoint family of open balls B : B ∈ F has to

be at most countable (why?).Let us observe that,

(2.4) for fixed B ∈ Gj ∃ B1 ∈ ∪ji=1Fi such that B1 ∩B 6= ∅ .Indeed, for if not, the family

F∗j = Fj ∪ B1would be a disjoint family of G∗j , thus contradicting the maximality of Fj. Moreover

(2.5) d(B) 6D

2j−1= 2

D

2j< 2 d(B1) ,

which implies that

(2.6) B ⊂ B1 .

Indeed, if B = B(x, r), B1 = B(x1, r1), let z ∈ B and, since (2.4), there existsy ∈ B ∩B1. Thus, by (2.5),

|z − x1| 6 |z − y|+ |y − x1| 6 d(B) + r1 6 2 d(B1) + r1 6 4 r1 + r1 = 5r1 .

and then (2.6) follows.

59

Remark 2.6. (i) We emphasize that the point of the lemma is that the subcol-lection consists of countable disjoint elements, a very important considerationsince countable additivity plays a central role in measure theory.

(ii) If D = sup d(B) : B ∈ G = ∞, Vitali’s covering may fail. For instanceExercise: Let G := B(h) : h ∈ N, then prove that there is no a pairwise

disjoint subfamily F ⊂ G such that ∪B∈GB ⊂ ∪B∈FB.(iii) If the cover G is composed of open balls, the conclusion of Vitali’s covering

theorem still holds.

Now we are going to show, even though F could not cover the whole A, at least,it covers almost all of A with respect to the Lebesgue measure, that is, the so- calledVitali covering property holds for the Lebesgue measure.

Theorem 2.7 (Vitali covering property for the Lebesgue measure). Let G be a familyof closed balls in Rn, which is a fine cover of a (possibly non measurable) set A ⊂ Rn

in Rn. Then there exists a disjoint subfamily F ⊂ G, at most countable, such that

Ln (A \ ∪F) = 0 ,

where Ln denotes the n-dimensional Lebesgue outer measure.

Proof. 1st step: Suppose A is bounded with 0 < Ln(A) < ∞, otherwise we aredone. Since Ln is a Borel regular outer measure, by Corollary 1.11, there is an openset U0 ⊂ Rn such that U0 ⊃ A and

(2.7) Ln(U0) 6 (1 + 7−n)Ln(A) .

Let

G0 := B ∈ G : B ⊂ U0, d(B) 6 1 .Being G fine, G0 is still a fine cover of A. Thus, by Vitali’s covering theorem 2.4, thereexists a disjoint, at most countable, subfamily F0 ⊂ G0 ⊂ G such that

A ⊂ ∪B∈GB ⊂ ∪B∈F0B .

Then

(2.8) 6−n Ln(A) < 5−n Ln(A) 6 5−n∑B∈F0

Ln(5B) =∑B∈F0

Ln(B) .

From (2.8), there exists a finite family of balls F1 := B1, . . . , Bk1 ⊂ F0 ⊂ G0 suchthat

(2.9) 6−n Ln(A) 6k1∑i=1

Ln(Bi) .

Define

A1 := A \ ∪F1 = A \ ∪k1i=1Bi .

If Ln(A1) = 0, we are done. Otherwise, from (2.7) and (2.9), we have that

Ln(A1) 6 Ln(U0 \ ∪k1i=1Bi) = Ln(U0)−k1∑i=1

Ln(Bi)

6 (1 + 7−n − 6−n)Ln(A) = uLn(A) ,

(2.10)

60

where 0 < u := 1 + 7−n − 6−n < 1. Now A1 ⊂ Rn \ (∪k1i=1Bi) and therefore we canfind an open set U1 such that

A1 ⊂ U1 ⊂ Rn \ (∪k1i=1Bi) ,

Ln(U1) 6 (1 + 7−n)Ln(A1) .

Arguing as above there are disjoint balls Bk1+1, . . . , Bk2 in G such that Bi ⊂ U1 fori = k1 + 1, . . . , k2 and, if

A2 := A1 \ ∪k2i=k1+1Bi = A \ ∪k2i=1Bi ,

(2.11) Ln(A2) 6 uLn(A1) .

Thus, from (2.10) and (2.11), it follows that there exists a finite family F2 :=B1, . . . , Bk2 ⊂ G of disjoint balls such that F1 ⊂ F2 and

Ln(A \ ∪F2) 6 u2 Ln(A) .

After m steps, we have that there exist m finite families Fi := B1, . . . , Bki ⊂ G(i = 1, . . . ,m) of disjoint balls such that F1 ⊂ · · · ⊂ Fm

(2.12) Ln(A \ ∪Fm) 6 um Ln(A) .

If Ln(A \ ∪Fm) = 0 for some m, the procedure stops and we are done. Otherwise weconstruct an increasing sequence (Fm)m of finite disjoint subfamilies of G such that(2.12) holds for each m ∈ N. Let us define

F := ∪∞m=1Fm .then, from (2.12), it follows that

Ln(A \ ∪F) 6 Ln(A \ ∪Fm) 6 um Ln(A) ∀m ∈ N .Taking the limit as m→∞ in the previous inequality, we complete the proof.

2nd step: Assume A unbounded. We can write Rn = ∪∞i=1Qi where (Qi)i is a

sequence of closed cubes of Rn such thatQi ∩

Qj = ∅ if i 6= j. Applying the first step

to A ∩Qi, for given i, and noticing that

Ln(A \ ∪∞i=1

Qi

)= 0

we complete the proof.

Remark 2.8. (i) A simple analysis of the proof of Theorem 2.7 yields that itstill holds true for a fine cover G composed of open balls.

(ii) All that we really used of the Lebesgue measure in the proof of Theorem 2.7was the equality Ln(B(x, 5r)) = 5n Ln(B(x, r)) in fact only the inequality ”6”. It is rather straightforward to modify the above proof to see that thetheorem remain valid if Ln is replaced by any Radon measure µ on Rn suchthat for some τ ∈ (1,∞),

(2.13) lim supr→0

µ(B(x, τr))

µ(B(x, r))< ∞ µ− a.e. x ∈ Rn .

61

Moreover, the balls can be replaced by more general families of closed setsand Rn by more general spaces, see Federer [Fe, 2.8] for example. However,the above theorem is not valid even for all very nice Radon measures on Rn,as the following example shows.

Example 2.9 (Vitali covering property does not hold for all Radon measures in Rn

[Ma]). Let µ be the Radon outer measure in R2 defined by

µ(E) := L1 (x ∈ R : (x, 0) ∈ E) if E ⊂ R2 .

It is easy to see that µ = H1 A where H1 denotes the 1-dimensional Hausdorffmeasure on R2 and A := (x, y) ∈ R2 : y = 0 . The family of balls

G := B((x, y), y) : 0 < y < ∞

covers finely A. But for any countable subfamily F ⊂ G

µ(A ∩ (∪F)) = 0 .

Thus a Vitali covering property like the Lebesgue measure (see Theorem 2.7) cannothold for a general Radon measure in Rn.

Here A touches only the boundaries of the balls of G. By a slight modification, assuggested in [Ma], we could find a family G such that each point of A is an interiorpoint of arbitrarily small balls of G and yet the conclusion of Theorem 2.7 fails.

Exercise (suggested by R. Serapioni). Let µ be the the Radon measure as beforewith A := (x, 0) : x ∈ [0, 1].

Let G be the family of open balls

G := Ux,n : x ∈ [0, 1], n ∈ N

with

Ux,n := U

((x,

1

n

), rn

)and rn :=

1

n+ α e−n ,

where α ∈ (0, 1] to be fixed later.

(i) G is a fine cover of A;(ii) µ(A ∩ Ux,n) 6

√3α e−

n2 for each x ∈ [0, 1] and n;

(iii) for a given n, the number of disjoint balls Ux,n with x ∈ [0, 1] is at most n/2;(iv) let F ⊂ G be a disjoint, countable family , then, by (ii) and (iii),

µ (∪F) 6∞∑n=1

n

2

√3α e−

n2 6√α

∞∑n=1

n e−n2 <

1

2

for α small enough.

Thus, by (iv), for any disjoint subfamily F ⊂ G,

µ (A \ (∪F)) = µ(A)− µ (A ∩ (∪F)) > µ(A)− µ (∪F) >1

2.

However, if we should require that each point of A is the centre (in fact, not toofar from the centre would be enough) of arbitrarily small balls of G we would get theconclusion of Theorem 2.7. Next we shall develop a covering theorem of this type.

62

Besicovitch’s covering theorem and Radon measures on Rn.

Again we shall first introduce a theorem called Besicovitch’s covering theorem,which originated from Besicovitch [Be1] and [Be2].

Besicovitch’s covering theorem 2.10. There are integers P (n)and Q(n) dependingonly on n with the following properties. Let A be a bounded subset of Rn, and let Gbe a family of closed balls such that each point of A is the centre of some ball of G.

(i) There is a finite or countable subfamily F ⊂ G which covers A and every pointof Rn belongs to at most P (n) balls of F , that is,

χA 6∑B∈F

χB 6 P (n) .

(ii) There are subfamilies F1, . . . ,FQ(n) ⊂ G covering A such that each Fi is dis-joint, that is,

A ⊂ ∪Q(n)i=1 (∪Fi)

andB ∩B′ = ∅ for B,B′ ∈ Fi with B 6= B′ .

Proof. See [Ma, Theorem 2.7].

We can now easily establish a Vitali-type covering theorem for arbitrary Radonmeasures on Rn.

Theorem 2.11 (Vitali covering property for Radon measures). Let ϕ be a Radon o.m. in Rn, A ⊂ Rn (even not ϕ-measurable) and G a family of closed balls. Assumethat G is a cover of A and

(2.14) inf r : B(x, r) ∈ G = 0 ∀x ∈ A .Then there is a disjoint subfamily F ⊂ G, at most countable, such that

ϕ (A \ ∪F) = 0 .

Proof. 1st step: Suppose first A is bounded and we may assume 0 < ϕ(A) < ∞.By Theorem 1.14, there is an open set U0 such that A ⊂ U0 and

ϕ(U0) < (1 + (4Q(n))−1)ϕ(A) ,

where Q(n) is as in Besicovitch’s covering theorem 2.10 . By that theorem we canfind subfamilies F1, . . . ,FQ(n) ⊂ G such that each Fi is disjoint and

A ⊂ ∪Q(n)i=1 ∪ Fi ⊂ U0 .

Then

ϕ(A) 6Q(n)∑i=1

ϕ (∪Fi)

and consequently there is an i∗ with

ϕ(A) < Q(n)ϕ (∪Fi∗) .Further, for some finite subfamily F ′i∗ ⊂ Fi∗ we have

ϕ(A) < 2Q(n)ϕ(∪F ′i∗

).

63

Letting

A1 = A \ ∪F ′i∗ ,

we get with u = 1− 14Q(n)−1 < 1

ϕ(A1) < ϕ(U0 \ ∪F ′i∗

)= ϕ(U0)− ϕ

(∪F ′i∗

)<

(1 +

1

4Q(n)−1 − 1

2Q(n)−1

)ϕ(A) = uϕ(A)

We can now continue by the same principle as in the proof of Theorem 2.7.2nd step: Assume A unbounded. We may modify the last step of the proof of

Theorem 2.7 making use of the fact that, by Lemma 1.112, ϕ(H) can be positive forat most countably many parallel hyperplanes H. More precisely, if i = 1, . . . , n andt ∈ R, denote by

H(i)t := x = (x1, . . . , xn) ∈ Rn : xi = t ,

and

Ni :=t ∈ R : ϕ

(H

(i)t

)> 0

.

Then Ni is at most countable for each i = 1, . . . , n. Therefore, for each i = 1, . . . , n,

there exists a sequence (t(i)k )k∈Z ⊂ R such that

k 6 t(i)k < k +

1

2and ϕ

(H

(i)

t(i)k

)= 0 ∀ k ∈ Z .

Let us define the closed rectangles

Qk := Πni=1[t

(i)k−1, t

(i)k ] if k ∈ Z .

Then, it is easy to see that

Rn = ∪k∈ZQk,Qk ∩

Qk′ = ∅ if k 6= k′

and

ϕ

(Rn \ ∪k∈Z

Qk

)= 0 .

Applying the first step to A ∩Qk we complete the proof.

Remark 2.12. The above theorem still holds true for families of open balls if ϕ isthe Lebesgue measure (in this case it reduces to the classical Vitali covering theorem2.7); if ϕ is a general Radon measure, further conditions have to be imposed: forinstance, we may require that for every x ∈ A and ε > 0 the cardinality of the ballsof G centred at x with radius less than ε > 0 is more than countable, or that thisproperty fails for a ϕ-negligible set of points. In this case, in fact, it is possible toselect only those balls B such that ϕ(∂B) = 0 (thus getting again a fine cover) andapply Theorem 2.11 to the cover given by the closure of the selected balls. However.it is interesting to note that Theorem 2.11 does not hold in its full generality forfamilies of open balls, as the next example shows.

64

Example 2.13. Let Q = qi : i ∈ N and consider, respectively, the Radon (outer)measure ϕ on R and family of open intervals G defined as

ϕ :=∞∑i=1

1

2iδqi , G := (a, b) : a, b ∈ Q, a < b .

Then G can be also meant as a family of open balls in R, which finely covers Q. Onthe other hand, for each countable, disjoint subfamily F ⊂ G it follows that

ϕ(Q \ ∪F) > ϕ(∪U∈F∂U) > 0 .

Historical notes:([dG, I.3]) The most classical covering theorem in differentiationtheory is that of Vitali [Vitali2], which has traditionally been the tool to obtain theLebesgue differentiation theorem in Rn . In its original form the theorem of Vitalirefers to closed cubic intervals and the Lebesgue measure.

Later on Lebesgue [Le2] and others gave it a less rigid geometric form replacingcubes by other sets ”regular” with respect to cubes, keeping always the restrictionto the Lebesgue measure. This restriction originates in the type of proof of thetheorem, essentially that given by Banach [Ba], which requires that homothetic setshave comparable measures. Also Caratheodory’s proof [C] is based on this propertyalthough it is a little different.

Besicovitch [Be1, Be2] and A.P. Morse [Mor] were the first in obtaining similar cov-ering lemmas for more general measures in order to prove differentiability propertiesanalogous to that of the Lebesgue theorem.

2.2. Derivatives of Radon measures on Rn. Lebesgue-Besicovitch differen-tiation theorem for Radon measures on Rn. In this section the environmentmetric space will be X = Rn and M = B(Rn).

Notation: If x ∈ Rn, r > 0, ν and µ are positive Radon measures on Rn, then weinterpret

ν(B(x, r))

µ(B(x, r)):=

ν(B(x, r))

µ(B(x, r))if x ∈ spt(µ)

∞ if x ∈ Rn \ spt(µ);

if f ∈ L1loc(Rn, µ), A ⊂ Rn bounded with µ(A) > 0,

∫Af dµ :=

∫Af dµ

µ(A),

Derivatives of positive Radon measures.

Let ν and ν be a positive Radon measure on Rn and assume that ν << µ. Then,from the Radon-Nikodym theorem for Radon measures, there exists w ∈ L1

loc(Rn, µ),

w > 0 such that w =dν

dµ, that is,

ν(E) :=

∫E

w dµ ∀E ∈ B(Rn) .

Question:

(D) ∃ limr→0

ν(B(x, r))

µ(B(x, r))= lim

r→0

∫B(x,r)

w(y) dµ(y) = w(x) µ− a.e. x ∈ Rn ?

65

Definition 2.14. Let ν and µ be positive Radon measures on Rn.

(i) The upper and lower derivatives of ν with respect to µ at a point x ∈ Rn aredefined by

(2.15) Dµν(x) = lim supr→0

ν(B(x, r))

µ(B(x, r))∈ [0,+∞] ;

(2.16) Dµν(x) = lim infr→0

ν(B(x, r))

µ(B(x, r))∈ [0,+∞] .

(ii) At a point x ∈ Rn where the limit exists, we define the derivative of ν withrespect to µ by

(2.17) Dµν(x) = Dµν(x) = Dµν(x) .

The basic differentiation result for positive Radon measures on Rn is contained inthe following.

Theorem 2.15 (Differentation for positive Radon measures). Let ν and µ be positiveRadon measures on Rn.

(i) The derivative Dµν(x) exists and is finite (that is Dµν(x) ∈ [0,∞)) for µ-a.e.x ∈ Rn.

(ii) The function Dµν : Rn → [0,+∞] is Borel measurable, by defining Dµν = ∞on the possible µ-negligible set where it does not exist.

(iii) Let

(2.18) A := x ∈ Rn : ∃Dµν(x) ∈ [0,∞) .For all Borel sets B ⊂ Rn

(2.19)

∫B

Dµν dµ = ν(A ∩B) 6 ν(B) ,

with equality if ν << µ. In this case

Dµν(x) =dν

dµ(x) =

dνacdµ

(x) µ-a.e. x ∈ Rn .

denotingdλ

dµthe Radon-Nikodym derivative of λ with respect to µ.

(iv) ν << µ if and only if Dµν(x) < ∞ ν-a.e. x ∈ Rn.

As a corollary we obtain the following fundamental result.

Theorem 2.16 (Lebesgue-Besicovitch differentiation theorem). Let µ be a positiveRadon measure on Rn and let f ∈ L1

loc(Rn, µ). Then

∃ limr→0

∫B(x,r)

f(y) dµ(y) = f(x) µ− a.e. x ∈ Rn ,

that is, by definition, there exists a µ-negligible set N ⊂ Rn (i.e. µ(N) = 0) suchthat

(?) ∃ limr→0

∫B(x,r)

f(y) dµ(y) = f(x) ∀x ∈ Rn \N .

66

Remark 2.17. If f ∈ L1loc(Rnµ), denote by

[f ] :=g : Rn → R : g measurable, g = f µ- a.e. in Rn

.

Then the previous theorem actually states that there exists a µ-negligible set N(f)such that

∃ limr→0

∫B(x,r)

g(y) dµ(y) = f(x) ∀x ∈ Rn \N(f), g ∈ [f ] ,

and

f(x) = f(x) ∀x ∈ Rn \N(f) .

Thus the limit in (?) provides a way to define the value of f at x that is independentof the choice of representative in the equivalence class of f . Notice also that (?) canbe written as

∃ limr→0

∫B(x,r)

(f(y)− f(x)) dµ(y) = 0 µ− a.e. x ∈ Rn .

Lebesgue-Besicovitch differentiation theorem yields at once the following resultabout the density of a point with respect to a set.

Corollary 2.18 (Density of a set). Let µ be a positive Radon measure on Rn and letE ⊂ Rn be a measurable set. Then

∃ limr→0

µ(E ∩B(x, r))

µ(B(x, r))=

1 for µ-a.e. x ∈ E0 for µ-a.e. x ∈ Rn \ E

,

that is, µ-a.e. x ∈ E is a point of density 1 for E and µ-a.e. x ∈ Rn \ E is a pointof density 0 for E.

Proof of Theorem 2.16. It is not restrictive to assume that f > 0. Otherwise we candecompose f = f+ − f− and to apply the same argument to f+ and f−.

Let us define the Radon measure

ν(B) :=

∫B

f dµ if B ∈ B(Rn) .

By applying Theorem 2.15 (iii) we get that∫B

Dµν(x) dµ(x) = ν(B) =

∫B

f dµ(x) ∀B ∈ B(Rn) ,

whence

Dµν(x) = f(x) µ-a.e. x ∈ Rn .

Proof of Theorem 2.1. For each % ∈ Q, apply Theorem 2.16 to conclude that there isa set N% ⊂ Rn with µ(N%) = 0 such that

(2.20) limr→0

∫B(x,r)

|f(y)− %| dµ(y) = |f(x)− %| ∀x ∈ Rn \N% .

Thus, with N = ∪%∈QN%, we have µ(N) = 0. Moreover, for x ∈ Rn \N , % ∈ Q, since

|f(y)− f(x)| 6 |f(y)− %|+ |%− f(x)| ,

67

(2.20) implies that

lim supr→0

∫B(x,r)

|f(y)− f(x)| dµ(y) 6 2 |f(x)− %| ∀x ∈ Rn \N, % ∈ Q .

Since

inf |f(x)− %| : % ∈ Q = 0 ,

the proof is complete.

The proof of Theorem 2.15 relies on two preliminary results: the Vitali coveringproperty for Radon measures on Rn (see Theorem 2.11), which yields the followingcomparison between ν and µ provided a pointwise estimates on the derivatives of νwith respect to µ.

Lemma 2.19 (density estimates for differentiation of measures). Let ν and µ bepositive Radon measures on Rn, let A ⊂ Rn be a Borel set and 0 < t < ∞.

(i) If Dµν(x) 6 t for each x ∈ A, then ν(A) 6 t µ(A);

(ii) If Dµν(x) > t for each x ∈ A, then ν(A) > t µ(A);(iii) Dµν, Dµν : Rn → [0,∞] are Borel measurable.

In particular

(2.21) µ(x ∈ Rn : Dµν(x) = ∞) = 0 .

Proof. (i) It is not restrictive to assume that A is bounded. By the approximation ofBorel measures by open and closed sets (see Corollary 1.12), for every ε > 0 thereexists a bounded open set U ⊃ A such that µ(U) < µ(A)+ ε. Moreover, by definitionof lim inf, for each x ∈ A and ε > 0, there exists a sequence of positive real numbers(rh)h such that

(2.22) limh→∞

rh = 0 and ν(B(x, rh)) 6 (t+ ε)µ(B(x, rh)), B(x, rh) ⊂ U ∀h.

Fix ε > 0 and let us define

G :=B(x, r) : x ∈ A, r ∈ (0,+∞) with B(x, r) ⊂ U, ν(B(x, r)) 6 (t+ε)µ(B(x, r))

, .

From (2.22), it is easy to see that G is a family of closed balls which covers A andsatisfies (2.14). Applying the Vitali covering property for Radon measures on Rn (seeTheorem 2.11), there exists a countable subfamily

F := Bi : i ∈ N ⊂ Gsuch that

Bi ∩Bj = ∅ if i 6= j and ν (A \ ∪∞i=1Bi) = 0 .

Thus

ν(A) 6 ν(∪∞i=1Bi) =∞∑i=1

ν(Bi) < (t+ ε)∞∑i=1

µ(Bi) = (t+ ε)µ(∪∞i=1Bi)(2.23)

6 (t+ ε)µ(U) < (t+ ε) (µ(A) + ε) .

Letting ε→ 0, the desired inequality follows.(ii) It is not restrictive to assume that A is bounded. By the approximation of

Borel measures by open and closed sets, for every ε > 0 there exists an open set

68

U ⊃ A such that ν(U) < ν(A)+ ε. Moreover, by definition of lim sup, for each x ∈ Athere exists a sequence of positive real numbers (rh)h such that

(2.24) limh→∞

rh = 0 and ν(B(x, rh)) > (t− ε)µ(B(x, rh)), B(x, rh) ⊂ U ∀h, .

Fix ε > 0 and let us define

G :=B(x, r) : x ∈ A, r ∈ (0,+∞) with B(x, r) ⊂ U, ν(B(x, r)) > (t−ε)µ(B(x, r))

.

From (2.24), it is easy to see that G is a family of closed balls which covers A andsatisfies (2.14). Applying the Vitali covering property for Radon measures on Rn (seeTheorem 2.11), there exists a countable subfamily

F := Bi : i ∈ N ⊂ G .such that

(2.25) Bi ∩Bj = ∅ if i 6= j and µ (A \ ∪∞i=1Bi) = 0 .

Thus, by (2.25) and (2.24),

(t− ε)µ(A) 6 (t− ε)µ(∪∞i=1Bi) =∞∑i=1

(t− ε)µ(Bi) 6∞∑i=1

ν(Bi)(2.26)

= ν(∪∞i=1Bi) 6 ν(U) < (ν(A) + ε) .

Letting ε→ 0, the desired inequality follows.(iii) Fix r > 0 and let gr : Rn → [0,∞] be the function

gr(x) :=ν(B(x, r))

µ(B(x, r))if x ∈ Rn .

Exercise: Prove that:(a) given r > 0, the function Rn 3 x 7→ ν(B(x, r)) (and also the function Rn 3

x 7→ µ(B(x, r))) is upper semicontinuous, that is

ν(B(x, r)) > lim supy→x

ν(B(y, r)) ∀x ∈ Rn ;

(b) given r > 0, the function gr : Rn → [0,∞] is Borel measurable;(c) given x ∈ Rn, the function (0,∞) 3 r 7→ ν(B(x, r)) (and also the function

(0,∞) 3 r 7→ µ(B(x, r))) is right-continuous, that is

lims→r+

ν(B(x, s)) = ν(B(x, r)) ∀ r ∈ (0,∞) ;

(d) Dµν(x) = limh→∞

(inf

r∈(0,1/h)∩Qgr(x)

), Dµν(x) = limh→∞

(sup

r∈(0,1/h)∩Qgr(x)

)for

each x ∈ Rn.From (d) and (b), it follows that both Dµ and Dµ are Borel measurable.Let us now prove (2.21). For each h ∈ N let

Eh := x ∈ B(h) : Dµν(x) = ∞ .By previous claim (ii), it follows that, for fixed h,

µ(Eh) 6ν(B(h))

t∀ t ∈ [1,∞) .

69

Since ν(B(h)) <∞, taking the limit as t→∞ in the previoua inequality, we get that

µ(Eh) = 0 ∀h ∈ Nand then (2.21) follows.

Proof of Theorem 2.15. (i) For 0 < R < ∞, 0 < s < t < ∞, let

As,t,R :=x ∈ B(0, R) : Dµν(x) 6 s < t < Dµν(x)

,

At,R :=x ∈ B(0, R) : t 6 Dµν(x)

.

Then As,t,R and At,R are Borel sets and, by Lemma 2.19,

t µ(As,t,R) 6 ν(As,t,R) 6 s µ(As,t,R) < ∞ ,

u µ(Au,R) 6 ν(Au,R) 6 ν(B(0, R)) < ∞ .

These inequalities yield

(2.27) µ(As,t,R) = 0 ∀ 0 < s < t < ∞ ,

and

(2.28) µ(∩u>0Au,R) = limu→∞

µ(Au,R) = 0 .

Notice that∩u>0Au,R = ∩n∈NAn,R ,

and the ∩u>0Au,R is a Borel set.Let denote by N1 the set of points x ∈ Rn such that @Dµν(x) or Dµν(x) = ∞.Exercise: Prove that

N1 = ∪ As,t,R : 0 < s < t < ∞, s, t ∈ Q, R > 0, R ∈ Q ∪ ∩u>0Au,R : R > 0, R ∈ Q .

Then, from (2.27) and (2.28), it follows that µ(N1) = 0, which settles (i).(ii) By Lemma 2.19 (iii), both Dµν and Dµν are Borel measurable functions. Thus,

by definition, Dµν(x) is a Borel measurable function, too.(iii) Let A be the set defined in (2.18) and let N2 := x ∈ Rn : ∃Dµν(x) = 0.

Observe that

(2.29) Rn \ A ⊆ N1 .

Let us begin to prove that

(2.30) ν(N2) = 0 .

For 0 < R < ∞, 0 < s < ∞, let

A∗s,R := x ∈ B(0, R) : Dµν(x) 6 s .Then, by Lemma 2.19 (i), for given R > 0, it follows that

ν(N2 ∩B(0, R)) 6 ν(A∗ε,R) 6 ε µ(A∗ε,R) 6 ε µ(B(0, R)) ∀ ε > 0 .

Letting ε→ 0 in the previous inequality, we get

ν(N2 ∩B(0, R)) = 0 ∀R > 0 ,

which establishes (2.30). Let us now prove the identity in (2.19). This amounts toprove the two inequalities

70

(2.31)

∫B

Dµν(x) dµ(x) 6 ν(A ∩B) ∀B ∈ B(Rn) ,

(2.32)

∫B

Dµν(x) dµ(x) > ν(A ∩B) ∀B ∈ B(Rn) .

Fix B ∈ B(Rn) and choose 1 < t < ∞, let

Bk :=x ∈ B : tk 6 Dµν(x) < tk+1

k ∈ Z ,

and notice that

(2.33) (A \N2) ∩B = ∪k∈ZBk Bk ∩Bk′ = ∅ if k 6= k′ ,

where A is the set defined in (2.18). By (2.21), (2.28), Lemma 2.19 (i) and (2.30),∫B

Dµν(x) dµ(x) =

∫A∩B

Dµν(x) dµ(x) =∞∑

k=−∞

∫Bk

Dµν(x) dµ(x) 6∞∑

k=−∞

tk+1 µ(Bk)

6 t∞∑

k=−∞

ν(Bk) 6 t ν((A \N2) ∩B) = t ν(A ∩B) .

Letting t → 1+ in the previous inequality, we establish (2.31). Let us now show(2.32). Choose 0 < t < 1, let

Bk :=x ∈ B : tk+1 6 Dµν(x) < tk

k ∈ Z ,

an notice that (2.33) still holds. Arguing as before, we get∫B

Dµν(x) dµ(x) =

∫A∩B

Dµν(x) dµ(x) =∞∑

k=−∞

∫Bk

Dµν(x) dx >∞∑

k=−∞

tk+1 µ(Bk)

> t∞∑

k=−∞

ν(Bk) = t ν((A \N2) ∩B) = t ν(A ∩B) .

Letting t → 1−, in the previous inequality, we get (2.32) and then the equality in(2.19). The inequality in (2.19) trivially follows by the monotonicity of ν.

If ν << µ, we have to prove that

ν(A ∩B) = ν(B) ∀B ∈ B(Rn) .

We need only to show that

ν(Rn \ A) = 0 .

By (2.29), we have only to prove that

(2.34) ν(N1) = 0 .

By previous point (i), and because of ν << µ, (2.34) follows.(iv) If ν << µ, by previous claim (i), Dµν(x) µ-a.e. x ∈ Rn and then Dµν(x) ν-a.e.

x ∈ Rn, too. Suppose now that Dµν(x) < ∞ ν-a.e. x ∈ Rn and let B ∈ B(Rn) withµ(B) = 0. Lemma 2.19 (i) gives

ν(x ∈ B : Dµν(x) 6 h

)6 hµ(B) = 0 ∀h ∈ N .

71

Therefore, since ν(x ∈ Rn : Dµν(x) = ∞) = 0,

ν(B) = ν(∪∞h=1

x ∈ B : Dµν(x) 6 h

)6

∞∑h=1

ν(x ∈ B : Dµν(x) 6 h

)= 0 .

We are now going to deal with the problem of the Lebesgue decomposition ofpositive Radon measure ν with respect to a given Radon measure µ, by characterizingthe absolutely continuous and singular parts in terms of derivatives of measures.

From Theorem 2.15, the following result immediatly follows.

Theorem 2.20 (Lebesgue decomposition in terms of derivatives of measures). Let νand µ be positive Radon measures on Rn. Let νac and νs denote respectively the abso-lutely continuous and singular parts of ν in the Lebesgue decomposition with respectto µ. Then, for each Borel set B,

νac(B) =

∫B

Dµν(x) dµ, νs(B) = ν S(B) = ν(S ∩B)

where S is the µ-negligible Borel set

S = (Rn \ spt(µ)) ∪ x ∈ spt(µ) : Dµν(x) = ∞

Proof. Let us recall that, by definition, Dµν(x) = ∞ either if @Dµν(x) or if x /∈ spt(µ)and that A ⊂ spt(µ), where A is the set defined in (2.18). Thus, by Theorem 2.15(i), µ(S) = 0 and

Rn \ S = A .

Moreover we can write

(2.35) ν = ν A+ ν S

According to (RN) and by (2.19),

(2.36) ν A = µw << µ with w = Dµν ,

and, since ν A and ν S are mutually singular, by (2.35), (2.36)and the uniquenessof Lebesgue decomposition (see Theorem 1.32), we get the desired conclusion.

As a byproduct of the previous theorem we get the following result (see also [SC,Lemma 1.51]).

Corollary 2.21. Let ν and µ be two positive Radon measures on Rn that are (mutu-ally) singular, that is, ν ⊥ µ. Then

∃Dµν(x) = 0 µ-a.e. x ∈ Rn .

Proof. From Lebesgue decomposition theorem 1.32, it follows that νac ≡ 0 and ν = νs.Thus, from Theorem 2.20, it follows that

0 = νac(B) =

∫B

Dµν dµ for each B ∈ B(Rn) .

This implies that Dµν(x) = 0 µ-a.e. x ∈ Rn.

72

In several applications of the differentiation of measures, a differentiation resultwith respect to a more general class of sets rather than balls could very useful as,for instance, the case of the Lebesgue differentiation theorem for monotone functions(see [GZ, SC]). Therefore we are going to show that a more general class of sets canbe considered in the derivative of measures, instead of balls.

Definition 2.22. Let µ be a positive Radon measure on Rn and let x ∈ Rn. Asequence of Borel sets (Eh(x))h ⊂ B(Rn) is called a (regular) differentiation basis at

x for µ provided there is αx = α(x) > 0 with the following properties:

(i) there exists a sequence of balls (B(x, rh))h with rh → 0 such that

Eh(x) ⊂ B(x, rh) ∀h ;

(ii) µ(Eh(x)) > αx µ(B(x, rh)) ∀h .

Theorem 2.23 (Differentiation for positive Radon measures with respect to a dif-ferentiation basis). Let ν and µ be postive Radon measures on Rn, then there exists

limh→∞

ν(Eh(x))

µ(Eh(x))=

dνacdµ

(x) = Dµνac(x) µ-a.e. x ∈ spt(µ) ,

whenever (Eh(x))h is a differentiation basis of µ at x.

Proof. Recall that, by Corollary 2.21, Theorems 2.20 and 2.15 (i),

(2.37) ν = νac + νs

with νac << µ and νs ⊥ µ,

(2.38) ∃ limr→0

νs(B(x, r))

µ(B(x, r))= 0 µ-a.e. x ∈ spt(µ) ,

(2.39) ∃ limr→0

ν(B(x, r))

µ(B(x, r))= lim

r→0

νac(B(x, r))

µ(B(x, r))=

dνacdµ

(x) := w(x)µ-a.e. x ∈ spt(µ) .

Let x ∈ spt(µ) be such that (2.38) holds and let (Eh(x))h be a differentiation basisat x of µ. Then

0 6νs(Eh(x))

µ(Eh(x))6

1

αx

νs(B(x, rh))

µ(B(x, rh)),

and, from (2.38), it follows that

(2.40) ∃ limh→∞

νs(Eh(x))

µ(Eh(x))= 0 .

To conclude the proof we need only to prove that

(2.41) ∃ limh→∞

νac(Eh(x))

µ(Eh(x))= w(x)

when x ∈ spt(µ) is a Lebesgue point of w (see Definition 2.2).Observe that such apoint x satisfies (2.39), too. Thus, (2.37), (2.40) and (2.41) conclude the proof. FromLebesgue points theorem,

∃ limh→∞

∫B(x,rh)

|w(y)− w(x)| dµ(y) = 0 µ-a.e. x ∈ spt(µ) .

73

Now observe that∫Eh(x)

|w(y)− w(x)| dµ(y) =1

µ(Eh(x))

∫Eh(x)

|w(y)− w(x)| dµ(y) 6

µ(B(x, rh))

µ(Eh(x))

∫B(x,rh)

|w(y)− w(x)| dµ(y) 61

αx

∫B(x,rh)

|w(y)− w(x)| dµ(y)→ 0 .

Thus, it follows that∣∣∣∣ νac(Eh(x))

µ(Eh(x))− w(x)

∣∣∣∣ =

∣∣∣∣ ∫Eh(x)

w(y) dµ(y)− w(x)

∣∣∣∣ =∣∣∣∣ ∫Eh(x)

(w(y)− w(x)) dµ(y)

∣∣∣∣ 6 ∫Eh(x)

|w(y)− w(x)| dyµ(y)→ 0 .

Then (2.41) holds.

Derivatives of vector Radon measures.

We are now going to deal with the problem of the Lebesgue decomposition of vectorRadon measure ν with respect to a given Radon measure µ, by characterizing theabsolutely continuous and singular parts in terms of derivatives of measures.

Theorem 2.24. Let ν and µ be respectively a Rm-valued Radon and a positive Radonmeasures on Rn. Then, for µ-a.e. x ∈ spt(µ),

(2.42) ∃w(x) := limr→0

ν(B(x, r))

µ(B(x, r))∈ Rm .

Moreover the Lebsesgue decomposition of ν with respect to µ is given by ν = µw + νswhere νs(B) = ν S(B) = ν(S ∩B) and S is the µ-negligible Borel set

S = (Rn \ spt(µ)) ∪ x ∈ spt(µ) : Dµ|ν|(x) = ∞

Proof. Let us recall that, by the polar decomposition of ν with respect to |ν| (seeCorollary 1.53) and the Lebsegue decomposition in terms of derivatives of measures(see Theorem 2.20)

(2.43) ν(B) =

∫B

wν d|ν| ∀B ∈ B(Rn) ,

(2.44) |ν|(B) =

∫B

Dµ|ν| dµ+ ν S(B) ∀B ∈ B(Rn) ,

where |ν| denotes the total variation of ν and wν : Rn → Rm is a Borel measurablevector function with |wν | = 1 |ν|-a.e. in Rn. By combining (2.43) and (2.44), itfollows that

(2.45) ν(B) =

∫B

Dµ|ν|wν dµ+

∫B

wν d(ν S) ∀B ∈ B(Rn) .

In particular, by (2.45) and the the uniqueness of the Lebesgue decomposition of νwith respect to µ (see Theorem 1.52), we get

(2.46) νac(B) =

∫B

Dµ|ν|wν dµ, νs(B) =

∫B

wν d(ν S) ∀B ∈ B(Rn) .

74

In addition, since S is µ-negligible, by (2.45) and the Lebesgue-Besicovitch differen-tiation theorem 2.16, we can infer

(2.47) ∃w(x) := limr→0

ν(B(x, r))

µ(B(x, r))= Dµ|ν|(x)wν(x) ∈ Rm µ a.e. x ∈ spt(µ) .

From (2.45), (2.46) and (2.47), we get the desired conclusion.

2.3. Extensions to metric spaces.

About Vitali’s covering theorem in metric spaces.

Vitali’s covering theorem still holds in any separable metric space (see, for instance,[AT, Theorem 2.2.3]).

If (X, d) is a metric space boundedly compact, that is all bounded and closed setsof X are compact, a constructive proof of Vitali’s covering theorem, whithout theHausdorff Maximal Principle, can be given (see [Ma, Theorem 2.1]).

If (X, d) is a metric space, using the Hausdorff maximal principle, one can givemuch more general Vitali’s covering type-results ; for example families of balls can bereplaced by many other families of sets (see, for instance, [Fe, 2.8.4-6]).

About Vitali’s covering property in metric measure spaces.

We call metric measure space a structure (X, d, µ) where (X, d) is a metric space andµ is a positive Radon measure on (X,B(X)). Given a metric measure space (X, d, µ)an important issue in the setting of analysis in metric space is to know whetherVitali’s covering property holds for the measure µ, that is, whether Theorem 2.11 stillholds replacing metric measure space (Rn, | · |, µ) with (X, d, µ). More precisely wesay that Vitali’s covering property holds for the metric measure space (X, d, µ) if, foreach A ⊂ X measurable and bounded, for each family of closed balls G covering Aand satisfying (2.14), there exists a disjoint subfamily F ⊂ G, at most countable,such that µ(A \ ∪F) = 0.

When speaking of a closed ball B in (X, d), it will be understood B that it comeswith a fixed center and radius (although these in general are not uniquely determinedby B as a set, since neither center nor radius need be unique). Thus B = B(x, r) forsome x ∈ X and some r > 0.

Case of the doubling spaces. Vital’s covering property holds when the measureµ is supposed to be doubling on metric space (X, d), that is we assume that:

• µ(X) > 0;• µ(B(x, r)) < ∞ ∀x ∈ X, r > 0;• there exists a positive constant C > 0 such that

(2.48) µ(B(x, 2r)) 6 C µ(B(x, r)) ∀x ∈ X, r > 0

(see, for instance, [He, Theorem 1.6]). Indeed condition (2.48) can be weakened inasymptotic doubling condition (2.13) (see [Fe, 2.8]).

Let us also recall that a metric space (X, d) is said to be doubling if there exists aninteger C > 1 such that each closed ball with radius r > 0 can be covered with lessthan C balls with radius r/2. If µ is a doubling measure on a metric space (X, d),

75

then it is easy to see that (X, d) is doubling. On the other hand, not every doublingspace carries a doubling measure (see [He, Chap.10]).

Case of the Besicovich covering property. Vitali covering property also holdsfor a metric measure space (X, d, µ) with metric space (X, d) satisfying the so-calledBesicovitch Covering Property (BCP) or also with (X, d) doubling metric space sat-isfying Weak Besicovitch Covering Property (WBCP) (see [LeR]).

Definition 2.25 (Besicovitch Covering Property). We say that BCP holds for (X, d)if there exists an integer P > 1 with the following property. Let A be a boundedsubset of (X, d) and let G be family of closed balls in (X, d) such that each point ofA is the center of some ball of G. Then there is subfamily F ⊆ G, at most countable,whose balls cover A and such that every point in X belongs to at most P balls of F ,that is,

χA 6∑B∈F

χB 6 P .

Definition 2.26 (Weak Besicovitch Covering Property). We say that WBCP holdsfor (X, d) if there exists an integer N > 1 with the following property. Suppose thereexist k points xi ∈ X and positive numbers ri > 0 (i = 1, . . . , k) such that

xi /∈ B(xj, rj) if i 6= j and ∩ki=1 B(xi, ri) 6= ∅ .

Then k 6 N .

The validity of BCP implies the one of WBCP. We stress that there exists metricspaces for which WBCP holds although BCP is not satisfied. However, when themetric is doubling, both covering properties turn out to be equivalent. An exaustivestudy of this topic is carried in [LeR].

An early study of metric spaces satisfying BCP was carried out by Federer [Fe,2.8.9]. In particular he proved that BCP holds in compact Riemannian manifoldsand in normed vector spaces of finite dimension. It is also well-known that BCP neednot hold in sub-Riemannian structures (see, for instance, [LeR]).

It is simple to show that, if a metric space (X, d) satisfies either a doubling orBCP condition, then (X, d) has finite topological dimension (in the sense of Lebesguecovering dimension) [LeR, 8.3].

It is also easy to prove that WBCP need not hold in an infinite dimensional space.For instance, let X be an infinite dimensional Hilbert space and let Bi := B(ei, 1)(i ∈ N), where ei : i ∈ N ⊂ X is a set of orthonormal vectors. Then ei /∈ Bj ifi 6= j and 0 ∈ ∩∞i=1Bi. Moreover there exist metric measure spaces (X, d, µ) with Xseparable Hilbert space and µ finite measure for which Vitali’s covering property doesnot hold [Ti].

About the Lebesgue differentiation theorem in metric measure spaces.

If µ is a locally finite Borel measure on a metric space (X, d), we say that theLebesgue differentiation theorem holds on metric measure space (X, d, µ) if

∃ limr→0

∫B(x,r)

f(y) dy = f(x) µ-a.e. x ∈ X, ∀ f ∈ L1loc(X,µ) .

76

where

L1loc(X,µ) :=

g : X → R : g Borel measurable ,∀x ∈ X ∃ rx > 0 such that∫

B(x,rx)

|f(y)| dµ(y) <∞.

It can be proved that the Lebesgue differentiation theorem holds for a metric measurespace (X, d, µ) provided that it satisfies Vitali’s covering property (see, for instance,[He, Remark 1.13]). In particular, by the previous arguments, the Lebegue differen-tiation theorem holds either if (X, d, µ) is doubling or (X, d) satisfies BCP.

An interesting characterization of metric measure spaces satisfying the Lebesguedifferentiation theorem is due to D. Preiss [Pre]. Following his terminology, if (X, d)is a metric space, we say that d is finite dimensional on a subset Y ⊂ X if there existC∗ ∈ [1,∞) and r∗ ∈ (0,∞] with the following property. Suppose there exist k pointsxi ∈ Y and positive numbers ri ∈ (0, r∗) (i = 1, . . . , k) such that

xi /∈ B(xj, rj) if i 6= j and ∩ki=1 B(xi, ri) 6= ∅ .Then k 6 C∗. We say d is σ-finite dimensional if X can be written as a countableunion of subsets on which d is finite dimensional. Note that WBCP holds on (X, d)if and only if d is finite dimensional on X for some constant C∗ ∈ [1,∞) and withr∗ =∞.

Theorem 2.27 ([Pre]). Let (X, d) be a complete separable metric space. The Lebesguedifferentiation theorem holds on (X, d) for all locally finite Borel measures if and onlyif d is σ-finite dimensional.

An interesting byproduct of the Lebesgue differentiation theorem is the following.

Proposition 2.28. Let µ and ν be two locally finite Borel measures on a metric space(X, d) and assume that

(i) the Lebesgue differentiation theorem holds for both measure spaces (X, d, µ)and (X, d, ν);

(ii) µ(B(x, r)) = ν(B(x, r)) for each x ∈ X, 0 < r < ∞.

Then µ(B) = ν(B) for each B ∈ B(Rn).

An outline of the proof of Proposition 2.28 will be proposed in the proof of Theorem3.30.

The Lebesgue differentation theorem need not hold in an infinite dimensional metricspace (see [Ti]).

3. An introduction to Hausdorff measures ([AFP, EG, Ma, Mag]).

Motivation: In this section we introduce Hausdorff measures and dimension formeasuring the metric size of quite general sets. They will be one of the basic meansfor studying geometric properties of sets and expressing results that these studies leadto. In particular they are very useful in this study because:

• they measure both regular submanifolds and not regular subsets of Rn, suchas fractal sets;• they do not depend on the parametrization of the submanifold.

77

3.1. Caratheodory’s construction and definition of Hausdorff measures ona metric space and their elementary properties; Hausdorff dimension. Themain idea is the construction of (outer) measures in Rn which allow to measuressubset of (roughly speaking) ”dimension” m < n. For instance, in R3, we are goingto define three measures H1, H2 and H3 such that

H1(S) = length(S) if S is a curve

H2(S) = area(S) if S is a surface

H3(S) = volume(S) if S is a ball

.

The basic definitions and first results on Hausdorff measures and dimension are dueto Caratheodory [C2] in 1914 and Hausdorff [Ha] in 1919 and they can be introducedin the framework of a general metric space (X, d).

We will start with a more general construction, called Caratheodory’s construction.It will yield also many other measures some of which will be briefly recalled.

Caratheodory’s construction

Let (X, d) be a metric space, F a family of subsets of X and ζ : F → [0,∞] agiven nonnegative evaluation set function. We make the following two assumptions.

(Ca1) For every δ ∈ (0,∞], there is a sequence (Ei)i ⊂ F such that X = ∪∞i=1Eiand d(Ei) 6 δ for each i ∈ N.

(Ca2) For every δ > 0, there is E ∈ F such that ζ(E) 6 δ and d(E) 6 δ.

For 0 < δ 6 ∞ and A ⊂ X we define

(3.1) ψδ(A) := inf

∞∑i=1

ζ(Ei) : A ⊂ ∪∞i=1Ei, d(Ei) 6 δ, Ei ∈ F

Remark 3.1. Assumption (Ca1) was only introduced to guarantee that such cover-ings always exist. The role of (Ca2) is to have ψδ(∅) = 0. It also allows us to usecoverings (Ei)i∈I with I finite or countable without changing value of ψδ(A).

Remark 3.2. It is easy to see that ψδ is monotonic and subadditive so that it isan outer measure. Usually it is highly non-additive and not a Borel measure (seeExercise below).

Evidently,

ψδ(A) 6 ψε(A) whenever 0 < ε < δ 6 ∞ .

hence we can define

ψ = ψ(F , ζ)

as follows

(3.2) ψ(A) := limδ→0

ψδ(A) = supδ>0

ψδ(A) for A ⊂ X .

The measure-theoretic behaviour of ψ is much better than that of ψδ.

Theorem 3.3. (i) ψ is a Borel outer measure.(ii) If F ⊂ B(X), then ψ is a Borel regular outer measure.

78

Proof. Proof. (i) The proof that ψ is an outer measure is straightforward and leftto the reader. To show that ψ is a Borel outer measure, we apply Caratheodorycriterion (see Theorem 1.5 (iii)). Let A, B ⊂ X with d(A,B) > 0. Choose δ with0 < δ < d(A,B)/2. If the sets E1, E2, · · · ∈ F cover A ∪ B and satisfy d(Ei) < δ,then none of them can meet both A and B. Hence

∞∑i=1

ζ(Ei) >∑

A∩Ei 6=∅

ζ(Ei) +∑

B∩Ei 6=∅

ζ(Ei)

> ψδ(A) + ψδ(B) .

Taking the infimum over all such coverings we have ψδ(A ∪ B) > ψδ(A) + ψδ(B).But the opposite inequality holds also as ψδ is an outer measure, and so ψδ(A∪B) =ψδ(A) + ψδ(B). Letting δ → 0, we obtain ψ(A ∪B) = ψ(A) + ψ(B) as required.

(ii) If A ⊂ X, choose for every i = 1, 2, . . . sets Ei,1, Ei,2, · · · ∈ F such that

A ⊂ ∪∞j=1Ei,j, d(Ei,j) < 1/i and∞∑j=1

ζ(Ei,j) < ψ1/i(A) + 1/i .

Then B := ∩∞i=1(∪∞j=1Ei,j) is a Borel set such that A ⊂ B and ψ(A) = ψ(B). Thusψ is a Borel regular outer measure.

Hausdorff measures

Let (X, d) be separable metric space, 0 6 s < ∞, and choose

(3.3) F := P(X) = E : E ⊂ X ,

(3.4) ζ(E) = ζs(E) := αs d(E)s

with the interpretations 00 = 1 and d(∅)s = 0, where αs is a geometric constantwhich depends only on s and the environment (X, d) and will be fixed later in theEuclidean context for normalization purposes of constants (see (3.11)).

Exercise: Prove that (Ca1) and (Ca2) are satisfied under assumptions (3.3) and(3.4).

The resulting measure ψ is called the s-dimensional Hausdorff measure and denotedby Hs . So

(3.5) Hs(A) := limδ→0Hsδ(A) = sup

δ> 0Hsδ(A) .

and Hsδ is the s-dimensional Hausdorff pre-measure defined by

(3.6) Hsδ(A) := inf

αs

∞∑i=1

d(Ei)s : A ⊂ ∪i=1Ei, d(Ei) 6 δ

where, δ ∈ (0,∞]. Let us observe that Hs

δ : P(X)→ [0,∞] is an outer measure butis not a Borel outer measure (see [Ma, Chap.4, Ex. 1]).

Exercise: Let X = Rn, 0 < s < ∞, 0 < δ 6 ∞. Prove that

79

(i) Hsδ : P(Rn) → [0,∞] is increasing, countably subadditive and Hs

δ(∅) = 0. Inparticular it is an outer measure.

(ii) Hsδ is neither additive nor a Borel measure. Indeed it can be proved that, if

n > 2, 0 6 s 6 1, 0 < δ < ∞ and U ≡ Uδ = U(0, δ/2) = U(δ/2), then

Hsδ(U) = Hs

δ(U) = Hsδ(∂U) .

(iii) Hsδ(A) 6 Hs

σ(A) for each 0 6 σ < δ 6 ∞, 0 6 s < ∞.

(Hint: (ii) 1st step: Prove that,

if n = 2 and s = 1 ,

then

(3.7) H1δ(∂U) = δ .

Indeed, inequalityH1δ(∂U) 6 δ

is trivial by the the definition of H1δ (recall that α1 = 1). The reverse inequality is

less trivial and the key point for its proof is the following property of a circumference:given m+ 1 points P1, . . . , Pm+1 ∈ ∂U with m > 3 satisfying

P1, . . . , Pm are distinct, P1 = Pm+1, |Pi − Pi+1| < δ ∀ i = 1, . . . ,m ,

thenm∑i=1

|Pi − Pi+1| > δ .

Therefore, by (3.7), since

δ = H1δ(∂U) 6 H1

δ(U) 6 δ ,

it follows that

(3.8) H1δ(∂U) = H1

δ(U) = δ .

Let σ ∈ (0, δ) and notice that, by (3.8) and the subadditivity of H1σ

σ = H1σ(Uσ) 6 H1

σ(U) .

Prove now that

(3.9) δ 6 lim infσ→δ−

H1σ(U) 6 H1

δ(U) 6 δ .

By (3.9) and (3.8), the conclusion of claim (ii) follows if n = 2 and s = 1.2nd step: Let

n > 3, s = 1 and Γ := x = (x1, . . . , xn) ∈ ∂U : x3 = · · · = xn = 0 .Then it follows that

H1δ(Γ) 6 H1

δ(∂U) 6 δ .

On the other hand, since Γ is isometric to the circle ∂U of R2, arguing as in the firststep, it follows that

H1δ(Γ) = δ .

Therefore it follows thatH1δ(∂U) = δ .

80

Analogously, arguing as in the remaining cases of the first step, the conclusion ofclaim (ii) follows if n > 2 and s = 1.

3rd step: Let

n > 2, 0 < s < 1 .

It is trivial that

Hsδ(∂U) 6 Hs

δ(U) 6 αs δs and Hs

δ(U) 6 αs δs .

Observe now that, for each non negative sequence of real numbers (ai)i,(∞∑i=1

ai

)s

6∞∑i=1

asi .

By the 2nd step, this implies that

δs =(H1δ(∂U)

)s6 α−1

s Hsδ(∂U) 6 α−1

s Hsδ(U) 6 δs ,

and

δs =(H1δ(U)

)s6 α−1

s Hsδ(U) 6 δs .

Thus the conclusion of claim (ii) follows if n > 2 and 0 < s < 1.)

Remark 3.4. Observe that pre-measure Hsδ does not fit our project to define a

measure which gives back the usual measure on submanifold of Rn. Indeed, letX = R2, s = 1 and Γ := (x, x sin(1/x)) : 0 < x 6 1/π. Then it is well-knownthat length(Γ) = ∞, but

H1δ(Γ) < ∞ ∀ 0 < δ < ∞ .

Remark 3.5. We can consider in the previous procedure, in place of F = P(X) ,the family F of all closed balls of (X, d) and still the evaluation function in (3.4).As before, we can define pre-measure Ssδ := ψδ (see (3.1)) and the resulting measureψ := limδ→0 ψδ (see (3.2)) is the so-called s-dimensional spherical Hausdorff denotedby Ss. Measures Hs and Ss can differ, but they are equivalent. Indeed it holds

(3.10) Hs(A) 6 Ss(A) 6 2sHs(A) ∀A ⊂ X .

The integral dimensional Hausdorff measures play a special role. Let us start froms = 0. It is easy to see that H0 agrees with the counting measure # on X defined inExample 1.2 (i) Next, for s = 1, H1 also has a concrete interpretation as a generalizedlength measure. In particular, for a rectifiable curve Γ in Rn, H1(Γ) can be shown toequal the length of Γ (see Theorem 3.25). For unrectifiable curves H1(Γ) = ∞.

More generally, if m is an integer, 1 < m < n, and S is a sufficiently regularm-dimensional surface in Rn (for example, a C1 submanifold), then the restrictionHm S gives the surface measure on S, for a suitable choice of αm, as we will see.This follows for example from the area formula, of which we will deal with in the nextsection.

For s = n in Rn ,

(3.11) Ln = Hn ,

81

by choosing the constant αn :=Ln(B(1))

2nin the definition of Hn. Observe that, by

(3.11), it follows that

(3.12) Hn(B(x, r)) = 2n αn rn ∀x ∈ Rn, r > 0 .

The proof of the equality (3.11) is rather complicated and based on the so-calledisodiametric inequality

Ln(A) 6 αn d(A)n for A ⊂ Rn

which we will deal with in the next section. But to see that Hn = cLn with somepositive and finite constant c is much easier and a proof is given in [Ma, 4.3] and wewill give an other proof in Theorem 3.30.

For any s > n, Hs turns out to be a trivial null measure on Rn. Indeed we willshow that Hs(Rn) = 0 (see Theorem 3.11).

We shall now derive some simple properties of Hausdorff measures in a generalseparable metric space (X, d).

Theorem 3.6. Let s ∈ [0,∞), αs > 0 and ζ(E) := αs d(E)s for E ⊂ X. If

(i) F = F ⊂ X : F closed or

(ii) F = U ⊂ X : F open ,then ψ(F , ζ) = Hs, where ψ(F , ζ) is the set function defined in (3.2).

Proof. Assume that (i) holds. It is clear that, by definition,

(3.13) Hs 6 ψ(F , ζ) .

In order to prove the reverse inequality, let us first observe that, given E ⊂ X, thenE ⊂ E and d(E) = d(E) if E denotes the closure of E. Assume that Hs(A) < ∞,otherwise we are done. Then, by definition, Hs

δ(A) < ∞, for each δ ∈ (0,∞].Therefore, for fixed δ ∈ (0,∞] and each ε > 0, there is a sequence of sets (Ei)i suchthat that A ⊂ ∪∞i=1Ei ⊂ ∪∞i=1Ei with d(Ei) = d(Ei) < δ, and

ψδ(F , ζ)(A) 6 αs

∞∑i=1

d(Ei)s = αs

∞∑i=1

d(Ei)s 6 Hs

δ(A) + ε .

Passing to the limit as δ → 0 in the previous inequality and since ε is arbitrary, weget the desired inequality. Assume now that (ii) holds. Let us first observe that, foreach E ⊂ X and σ > 0 , Iσ(E) := x ∈ X : d(x,E) < σ is an open set with

Iσ(E) ⊃ E and d(Iσ(E)) 6 d(E) + 2σ .

Again, (3.13) is immediate. Let us prove the reverse inequailty. As before, we canassume that, for fixed δ ∈ (0,∞] and each ε > 0, there is a sequence of sets (Ei)isuch that that A ⊂ ∪∞i=1Ei with d(Ei) < δ, and

(3.14) αs

∞∑i=1

d(Ei)s 6 Hs

δ(A) + ε .

82

For given i ∈ N and ε > 0, by the continuity of function [0,∞) 3 σ 7→ (d(Ei) + 2σ)s,there exists σi = σ(i, s, ε) > 0 such that

(3.15) (d(Ei) + 2σi)s < d(Ei)

s +ε

αs 2i.

Let Ui := Iσi(E) for i ∈ N. Therefore, by (3.14) and (3.15), it follows that

ψδ(F , ζ)(A) 6 αs

∞∑i=1

d(Ui)s 6 αs

∞∑i=1

(d(Ei) + 2σi)s

6 αs

∞∑i=1

d(Ei)s +

∞∑i=1

ε

2i6 Hs

δ(A) + 2ε .

Passing to the limit as δ → 0 in the previous inequality and since ε is arbitrary, weget the desired inequality.

Combining Theorems 3.6 and 3.3, we get

Corollary 3.7. Hs is a Borel regular outer measure.

Notice that usually Hs is not a Radon outer measure since it need not be locallyfinite. For example, if s < n every non-empty open set in Rn has non σ-finite Hs

measure. But taking anyHs-measurable set A in Rn withHs(A) < ∞, the restrictionHs A is a Radon measure by Theorem 1.92 (ii).

Often one is only interested in knowing which sets have null Hs-measure. For thisit is enough to use any of the pre-measures Hs

δ, for example Hs∞. In fact we do not

need any measure for defining the null Hs-measure sets.

Lemma 3.8 (Hs-null sets). Let A ⊂ X, 0 < s < ∞ and 0 < δ 6 ∞. Then thefollowing conditions are equivalent:

(i) Hs(A) = 0.(ii) Hs

δ(A) = 0.(iii) ∀ ε > 0 ∃E1, E2, · · · ⊂ X such that

A ⊂ ∪∞i=1Ei and∞∑i=1

d(Ei)s < ε .

Proof. Implications (i)⇒ (ii) ⇒ (iii) are trivial. Let us prove implication (iii) ⇒ (i).Without loss of generality we can assume that s > 0, otherwise the conclusion istrivial. By assumptions, it follows that

d(Ei) 6 δ(ε) := ε1/s ∀ i ∈ N .Therefore

Hsδ(ε)(A) 6 ε ∀ ε > 0.

Passing to the limit as ε→ 0 in the previous inequality, we get the desired conclusion.

We will now compare measures Hs among them.

Theorem 3.9. For 0 < s < t < ∞ and A ⊂ X,

(i) Hs(A) < ∞ implies Ht(A) = 0,(ii) Ht(A) > 0 implies Hs(A) = ∞.

83

Proof. (i) Let A ⊂ ∪∞i=1Ei with d(Ei) 6 δ and αs∑∞

i=1 d(Ei)s < Hs

δ(A) + 1 < ∞ .Then

Htδ(A) < αt

∞∑i=1

d(Ei)t = αt

∞∑i=1

d(Ei)t−sd(Ei)

s

< αt δt−s

∞∑i=1

d(Ei)s 6

αtαsδt−s (Hs

δ(A) + 1) ,

which gives (i) as δ → 0.(ii) This claim is only a restatement of (i).

Hausdorff dimension

Theorem 3.9 enables the definition of an important notion in GMT, namely theHausdorff dimension. Very roughly, according to [Fa, Introduction], ” a dimensionprovides a description of how much space a set fills. It is a measure of the prominenceof the irregularities of a set when viewed at very small scales. A dimension containsmuch information about the geometrical properties of a set.”

Definition 3.10. Let (X, d) be a separable metric space, the Hausdorff (or alsometric) dimension of a set A ⊂ X

Hdim(A) = sup s : Hs(A) > 0 = sup s : Hs(A) =∞= inf

t : Ht(A) < ∞

= inf

t : Ht(A) = 0

where we put sup ∅ = 0 and inf ∅ = ∞ whether someone of the previous sets may beempty.

The Hausdorff dimension has the natural properties of monotonicity and stabilitywith respect to countable unions:

(3.16) Hdim(A) 6 Hdim(B) for A ⊂ B ⊂ X ,

(3.17) Hdim(∪∞i=1Ai) = supi

Hdim(Ai) for Ai ⊂ X, i = 1, 2, . . .

To state the definition in other words, Hdim(A) is the unique number (it may be ∞in some metric spaces) for which

(3.18) s < Hdim(A)⇒ Hs(A) = ∞, t > Hdim(A)⇒ Ht(A) = 0 .

Remark 3.11. At the borderline case s = Hdim(A) we cannot have any gen-eral nontrivial information about the value Hs(A); all three cases Hs(A) = 0,0 < Hs(A) < ∞, Hs(A) = ∞. If for some given A there is a s > 0 such that0 < Hs(A) < ∞, then s must equal Hdim(A).

Remark 3.12. Since, we will see in the next section that Hdim(Rn) = n (see Corol-lary 3.29). Hence 0 6 Hdim(A) 6 n for all A ⊂ Rn. One can prove that, for alls ∈ [0, n], Hdim(A) = s for some subset A of Rn (see Remark 3.35).

84

3.2. Recalls of some fundamental results on Lipschitz functions betweenEuclidean spaces and relationships with Hausdorff measures.

Recalls of some fundamental results on Lipschitz functions between Euclidean spaces.

Let us recall the general definition of Lipschitz function, which makes sense evenfor functions acting between metric spaces.

Definition 3.13. Let (X, d) and (Y, %) be metric spaces, let E ⊂ X and let f : E ⊂X → Y .

(i) f is said to be Lipschitz or L-Lipschitz if there is L > 0

(3.19) %(f(x1), f(x2)) 6 Ld(x1, x2) ∀x1, x2 ∈ E .The smallest constant L such that (3.19) holds is called Lipschitz constant off and denoted

Lip(f, E) := sup

%(f(x1), f(x2))

d(x1, x2): x1, x2 ∈ E, x1 6= x2

∈ [0,∞) .

(ii) f is said to be locally Lipschitz if for any compact subset of E there is L > 0such that (3.19) holds for all x1, x2 in the compact.

An important issue in GMT and, more generally, in Analysis is the extension ofa C1 or Lipschitz function f : E ⊂ Rn → R to the whole Rn. Let us first recall afundamental result due to Whitney about C1 function.

Theorem 3.14 (Whitney’s extension theorem). Let C be a closed set in Rn. Letf : C → R and v : C → Rn be continuous functions. Define

R(x, y) :=f(x)− f(y)− v(y) · (x− y)

|x− y|∀x, y ∈ C, x 6= y .

Suppose that for all compact sets K ⊂ C

(3.20) limr→0

sup |R(x, y)| : x, y ∈ K, 0 < |x− y| < r = 0 .

Then there is f ∈ C1(Rn) such that f |C = f and ∇f |C = v.

Proof. The proof can be found in [EG, Sect. 6.5].

Remark 3.15. Let us observe that, by classical Taylor’s formula, condition (3.20)is satisfied if f ∈ C1(Rn). Thus Whiteney’s extension theorem is actually a charac-terization of the C1-extension of a function defined on a closed set, and is a partialconverse to Taylor’s formula.

A real valued L-Lipschitz function, defined on a subset E of a metric space (X, d),can always be extended to a L-Lipschitz function defined on the entire space X.

Theorem 3.16 (Mc Shane’s extension theorem [McS]). Let (X, d) be a metric space

and f : E ⊂ X → R be L-Lipschitz. Then there is f : X → R such that f |E = f

and f is L-Lipschitz.

Proof. The proof is not difficult and can be found in the original Mc Shane’s paper[McS] as well as in many textbooks devoted to analysis in metric spaces (see, forinstance, [AT, Theorem 3.1.2] and [He, He2]).

85

Let us recall that as an immediate consequence of Mc Shane’s extension theorem isthe following extension theorem for Lipschitz maps with target an Euclidean space.

Corollary 3.17. Let f : E → Rm, E ⊂ (X, d) be an L-Lipschitz function. Then

there exists an√mL-Lipschitz function f : X → Rm such that f |E = f .

Corollary 3.17 follows by applying Theorem 3.16 to the coordinate functions off . The multiplicative constant

√m in the corollary is in fact redundant, but this is

harder to prove.

Theorem 3.18 (Kirszbraun’s theorem). Let f : E → Rm, E ⊂ Rn, be an L-Lipschitz

function. Then there exists an L-Lipschitz function f : Rn → Rm such that f |E = f .

Proof. See, for instance, [He2, Theorem 2.5].

An other fundamental result concerning Lipschitz functions, acting between Eu-clidean spaces, concerns their differentiability a.e.

Theorem 3.19 (Rademacher’s theorem[Rad]). Let f : Rn → R be locally Lipschitz.Then f is differentiable (in classical sense) Ln-a.e., that is,

∃∇f(x) := (∂1f(x), . . . , ∂nf(x)) for Ln-a.e. x ∈ Rn

and

(3.21) limy→x

f(y)− f(x)− df(x)(y − x)

|y − x|= 0

where df(x) : Rn → R denotes the (linear) differential map of f at x defined by

df(x)(v) := ∇f(x) · v ∀ v ∈ Rn .

Moreover ∇f ∈ (L∞loc(Rn))n.

Proof. See, for instance, [EG, Theorem 2, Sect. 3.1.2].

Remark 3.20. Let us point out that, in the 1-dimensional case, i.e. n = 1, Rademacher’stheorem is an immediate consequence of the fundamental theorem of calculus, sincea locally Lipschitz function f : R → R is locally absolutely continuous, i.e. f |[a,b] ∈AC([a, b]) for each a, b ∈ R with a < b (see, for instance, [SC]).

Rademacher’s theorem trivially extends to locally Lipschitz functions f = (f1, . . . , fm) :Rn → Rm. In this case we get the existence, Ln-a.e. x ∈ Rn, of the Jacobian matrixof f at x , denoted Df(x) and defined by

(3.22) Df(x) :=

∂1f1(x) . . . ∂nf1(x). . . . . . . . .. . . . . . . . .

∂1fm(x) . . . ∂nfm(x)

m×n

.

Moreover (3.21) now holds with the (linear) differential map df(x) : Rn → Rm

df(x)(v) := Df(x) · v ∀ v ∈ Rn ,

where the previous product has to be meant as product between m×n matrix Df(x)and the (column) vector v.

86

Whitney’s theorem together with Rademacher’s theorem yield an approximationof a Lipschitz function, which states that it coincides, up to a small set, with a C1

function.

Theorem 3.21 (Approximation of Lipschitz functions). Let f : Rn → R be a Lips-chitz function. Then for each ε > 0 there is a g ∈ C1(Rn) such that

Ln (x : f(x) 6= g(x) ∪ x : ∇f(x) 6= ∇g(x)) < ε .

In addition, there is a positive constant c = c(n) such that

supRn|∇g| 6 c Lip(f) .

Proof. See [EG, Sect. 6.6.1].

Let us now stress this simple relationship between Hausdorff measures and Lipschitzmaps.

Theorem 3.22 (Hausdorff measures vs. Lipschitz maps). Let f : E ⊂ Rn → Rm bea Lipschitz map. Then

Hs(f(E)) 6 Lip(f)sHs(E) ∀ 0 6 s < ∞ .

In particularHdim(f(E)) 6 Hdim(E)

Proof. Let (Ei)i be a countable covering of E by sets with diameter less than δ. Then(f(Ei))i is a covering of f(E) with

diam(f(Fi)) 6 Lip(f)diam(Fi) 6 Lip(f)δ ∀ i .Exploiting the arbitrariness of (Ei)i in the following inequalities,

HsLip(f)δ(f(E)) 6 αs

∞∑i=1

diam(f(Ei))s 6 Lip(f)s αs

∞∑i=1

diam(Ei)s ,

we getHsLip(f)δ(f(E)) 6 Lip(f)sHs(E) .

We let δ → 0+ to get the desired inequality.

Remark 3.23. By Theorem 3.22, we find that Hausdorff measures are decreasedunder projection over an affine subspace of Rn. Indeed, if H is an affine subspace ofRn and f : Rn → Rn is the projection of Rn over H, then Lip(f) = 1. The samehappens, of course, if we project over a convex set.

Remark 3.24. We say that f : E ⊂ Rn → Rm (1 6 n 6 m ) is an isometry if|f(x)− f(y)| = |x− y| for every x, y ∈ E.

If s > 0, E ⊂ Rn, and f : E ⊂ Rn → Rm is an isometry, then Hs(f(E)) = Hs(E),as we may see either by applying Theorem 3.22 to f and to any extension g of f−1 withLip(g) 6 1, or by the area formula (IAF). In particular, if π is an n-dimensionalplane in Rm, then there exists an orthogonal injection P : Rn → Rm, such thatπ = P (Rn), that is, there exists P : Rn → Rm injective and satisfying

(P (x), P (y))Rm = (x, y)Rn ∀x, y ∈ Rn and P (Rn) = π .

87

Indeed, let v1, . . . , vn ∈ π be a orthonormal basis of π with respect to the scalarproduct of Rm. Then

P (x1, . . . , xn) :=n∑i=1

xivi if (x1, . . . , xn) ∈ Rn

turns out to be the desired function. In particlar, notice that P is an isometry Itfollows that

Hn π = P#Hn .

On the left-hand side, Hn stands for the n-dimensional Hausdorff measure on Rm, onthe right-hand side, it denotes the n-dimensional Hausdorff measure on Rn (which inturn coincides with Ln; see Theorem 3.31). Indeed, notice that, if A ⊂ Rm,

π ∩ A = P(Rn ∩ P−1(A)

).

Since P : Rn → Rm is an isometry, we get

Hn(π ∩ A) = Hn(P(Rn ∩ P−1(A)

))= Hn

(Rn ∩ P−1(A)

)= P#Hn(A) .

Thus we get the desired identity.

3.3. Hausdorff measures in the Euclidean spaces; H1 and the classical no-tion of length in Rn; isodiametric inequality and identity Hn = Ln on Rn.

In the following of the section, we will assume that

X = Rn equipped with the Euclidean distance ,

and we will fix constant αs in (3.6) in such a way

(3.23) α∗s := 2s αs :=πs/2

Γ(s2

+ 1)

where Γ(t) :=∫∞

0e−t xt−1 dx if t > 0 is the Euler Gamma function.

Observe that, if s is equal to a positive integer n then α∗n agrees with the n-dimensional Lebesgue measure of a unit ball in Rn.

We are going to study here the further properties of Hausdorff measures taking thestructure of Rn into account.

First, Hausdorff measures behave nicely under translations and dilations in Rn :for A ⊂ Rn, a ∈ Rn, 0 < t <∞,

(3.24) Hs(A+ a) = Hs(A) where A+ a := x+ a : x ∈ A,

(3.25) Hs(tA) = tsHs(A) where t A := tx : x ∈ A .

These are readily verified from the definition.

Hausdorff measures and length measure

A curve of Rn is a continuous function γ : [a, b] → Rn; its support is the setγ([a, b]) = Γ ⊂ Rn and, in this case, γ is called a parametrization of Γ. For the sakeof simplicity, we can assume that a = 0 and b = a. Given a curve γ : [0, a] → Rn

88

and a subinterval [c, d] ⊆ [0, a], we define the length (or also variation) of γ over [c, d]as(3.26)

l(γ; [c, d]) := sup

m∑i=1

|γ(ti)− γ(ti−1)| : t0 = c < t1 < · · · < tm = d

∈ [0,∞]

where the supremum is taken over all finite partitions t0 = c < t1 < · · · < tm = dof [c, d]. Moreoever let us denote

l(γ) := l(γ; [a, b]) .

Exercise:

(i) l(γ; [b, c]) > |γ(b)− γ(c)|, whenever 0 6 b 6 c 6 a;(ii) l(γ; [b, c]) = l(γ; [b, d]) + l(γ; [d, c]), whenever 0 6 b 6 d 6 c 6 a.

It is also well-known that, if γ : [0, a]→ Rn is of class C1, then

(3.27) l(γ; [c, d]) =

∫ d

c

|γ′(t)| dt ∀[c, d] ⊂ [0, a] .

If l(γ; [0, a]) <∞, the curve γ is said to be rectifiable. Whether l(γ; [0, a]) is finite ornot, the following theorem holds true.

Theorem 3.25 (Classical length and H1). Let γ : [0, a]→ Rn be a curve and denoteΓ = γ([0, a]) its support. Then

H1(Γ) 6 l(γ)

and equality holds if γ : [0, a] ⊂ R→ Rn is injective.

Before the proof we need the following preliminary result and thanks to G.P.Leonardi for usefuel suggestions for the proof.

Lemma 3.26. Let γ : [0, a]→ Rn be a rectifiable curve, that is l := l(γ; [0, a]) < ∞.Let v : [0, a]→ [0, l] be the function defined by

(3.28) v(t) := l(γ; [0, t]) t ∈ [0, a] .

Then v is a non decreasing continuous function. In particular v([0, a]) = [0, l].

Proof. The feature that v is non decreasing is straightforward. Let us prove that v iscontinuous. Since v is non decreasing, we have only to prove that

lims→t+

v(s) = lims→t−

v(s) = v(t) ∀ t ∈ [0, a] .

Let us prove that

(3.29) lims→t−

v(s) = v(t) ∀ t ∈ (0, a] .

Since l(γ; [0, a]) < ∞, for each ε > 0 there exists a partition of [0, a], t0 = 0 < t1 <· · · < tm = a, such that

(3.30) l(γ; [0, a])−m∑i=0

|γ(ti)− γ(ti−1)| < ε .

89

Moreover, without loss of generality, we can suppose that ti0 = t for some i0 =1, . . . ,m. Otherwise, since there exists i0 = 1, . . . ,m such that ti0−1 < t < ti0 , wecould enlarge the previuos partition by defining a new partition

t∗i :=

ti if 0 6 i 6 i0 − 1

t if i = i0ti+1 if i0 6 i 6 m

and, because ofm∑i=0

|γ(ti)− γ(ti−1)| 6m+1∑i=0

|γ(t∗i )− γ(t∗i−1)| ,

we still have

l(γ; [0, a])−m+1∑i=0

|γ(t∗i )− γ(t∗i−1)|

6 l(γ; [0, a])−m∑i=0

|γ(ti)− γ(ti−1)| < ε .

Observe now, by claim (ii) of the previous exercise, we can write (3.30) as

l(γ; [0, a])−m∑i=0

|γ(ti)− γ(ti−1)|

=m∑i=0

(l(γ; [ti−1, ti])− |γ(ti)− γ(ti−1)|) < ε .

which implies,

(3.31) l(γ; [ti0−1, t])− |γ(t)− γ(ti0−1)| < ε .

Meanwhile, from (3.31), it follows that, for each ti0−1 6 s 6 t,

l(γ; [ti0−1, t])− (|γ(ti0−1)− γ(s)|+ |γ(s)− |γ(t)|)= l(γ; [ti0−1, s]) + l(γ; [s, t])− (|γ(ti0−1)− γ(s)|+ |γ(s)− |γ(t)|)l(γ; [ti0−1, t])− |γ(t)− γ(ti0−1)| < ε ,

which implies, for each ti0−1 6 s 6 t,

(3.32) v(t)− v(s)− |γ(s)− γ(t)| = l(γ; [s, t])− |γ(s)− γ(t)| < ε .

On the other hand, since γ is continuous, for each ε > 0 there is t = t(ε) ∈ (ti0−1, t)such that for each t < s < t

(3.33) |γ(t)− γ(s)| < ε .

Therefore, by (3.32) and (3.33), it follows that, for each ε > 0 there is t < t such that

v(t)− v(s) < 2ε ∀ s ∈ (t, t) ,

and (3.29) follows. To prove the right continuity, that is

(3.34) lims→t+

v(s) = v(t) ∀ t ∈ [0, a)

90

we can use the same procedure. Indeed, by (3.30), we can now suppose that ti0−1 = tfor some i0 = 1, . . . ,m. Arguing as before, we get that, for each t 6 s 6 ti0 ,

v(s)− v(t)− |γ(s)− γ(t)| = l(γ; [t, s])− |γ(s)− γ(t)| < ε

and then, by the continuity of γ, (3.34) follows.

Proof of Theorem 3.25. We will follow the proof in [Mag, Theorem 3.8]. Let us beginwith the following exercise.

Exercise: Prove that H1 = L1 on R, as outer measures.(Hint: Use the fact that L1(E) 6 d(E) for each E ⊂ R.)The theorem is proved by Remark 3.24 and the previous exercise if Γ is a segment,

Indeed, assume that Γ = γ([0, a]) with

γ(t) := p+ tp− q|p− q|

if 0 6 t 6 a := |p− q|

where p, q ∈ Rn with p 6= q. Since γ : [0, a] ⊂ R→ Rn is an isometry, then

H1(Γ) = H1 (γ([0, a]) = H1([0, a])

= L1([0, a]) = |p− q| = l(γ; [0, a]) .

Set l = l(γ; [0, a]). We divide the proof into three steps.1st step. Let us show that H1(Γ) > |γ(a)− γ(0)|. Since the projection p : Rn →

Rn of Rn onto the line defined by γ(0) and γ(a) satisfies Lip(p) 6 1, by Theorem3.22 we have

H1(p(Γ)) 6 H1(Γ) .

At the same time, p(Γ) must contain the segment S := tγ(a) + (1− t)γ(0) : 0 6t 6 1. Otherwise, Γ = γ([0, a]) would be disconnected, against the continuity of γ.Thus H1(p(Γ)) > H1(S) = |γ(a)− γ(0)|.

2nd step. Let us prove that H1(Γ) 6 l. If l = ∞, we are done. Thus we canassume that l < ∞ and we are going to construct a Lipschitz function γ∗ : [0, l]→ Rn

with Lip(γ∗) 6 1 and Γ = γ∗([0, l]). Indeed, by Theorem 3.22, the existence of γ∗

will imply, as required, that

H1(Γ) = H1(γ∗([0, l]) 6 H1([0, l]) = l .

First let us assume that γ is injective. To construct γ∗ (which is just the parametriza-tion by arc length of γ, defined without using derivatives), we define Then v(0) = 0,v(a) = l and v is strictly increasing, that is, v(t) < v(s) if t < s, as γ is injective. Inparticular, v is continuous and invertible, with a continuous strictly increasing inversew : [0, l]→ [0, a]. Let then γ∗ : [0, l]→ Rn be defined by

γ∗(s) := γ(w(s)) s ∈ [0, l] .

Then


(3.35) l(γ∗; [0, s]) = s ∀ s ∈ [0, l]

(Hint: prove that l(γ∗; [0, s]) = l(γ; [0, w(s)]) = v(w(s)) = s).

91

We easily find that Lip(γ∗) 6 1, since, by properties (i) and (ii) in the exerciseabove, if [s1, s2] ⊂ [0, l], then

|γ∗(s1)− γ∗(s2)| 6 l(γ∗; [s1, s2]) = l(γ∗; [0, s2])− l(γ∗; [0, s2]) = s2 − s1 .

If γ is not injective, even if the construction is more difficult (see, for instance,[AT,Theorem 4.2.1]), we can still construct a parametrization of Γ, γ∗ : [0, l] → Rn withLip(γ∗) 6 1 and we can argue as before.

3rd step. Suppose now that γ is injective.If t0 = 0, . . . , tm = a is a competitorin the definition of l, then, setting Γh := γ([th−1, th]) (h = 1, . . . ,m), we have Γ =∪mh=1Γh and, by the injectivity of γ, H1(Γh ∩ Γh+1) = H1(γ(th) = 0. We thus findH1(Γ) > l as, by step one,

H1(Γ) =m∑h=1

H1(Γh) >m∑h=1

|γ(th)− γ(th−1)| .

Remark 3.27. When γ : [0, a]→ Rn is of class C1, it is immediately seen that (3.27)holds with c = 0 and d = a. In particular, by Theorem 3.25, if γ is injective andΓ = γ([0, a]),

H1(Γ) =

∫ a

0

|γ′(t)| dt .

This is the one-dimensional case of the area formula discussed in the previous section.

Hausdorff measures and Lebesgue measure

We are going to compare the outer measures Ln andHs on Rn. Let us first estimatethe values of Hs on the balls.

Proposition 3.28.

(3.36) Hs(B(x, r)) = c(s, n) rs x ∈ Rn, 0 < r < ∞with c(s, n) positive and finite constant only when s = n; for s > n, c(s, n) = 0; fors < n, c(s, n) = ∞.

Corollary 3.29. (i) Hs is a (non trivial) Radon measure on Rn if and only s =n.

(ii) Hdim(A) = n for each (nonemtpy) open set A ⊂ Rn. In particular Hdim(Rn) =n .

Proof. (i) Is is an immeditae consequence of Proposition 3.28.(ii) By Proposition 3.28 and (3.18), it follows that Hdim(U(x, r)) = n for each

x ∈ Rn and r > 0, where U(x, r) is an an open ball centered at x and with radiusr > 0. Indeed, for fixed r > 0, let (rh)h be a strictly increasing sequence of positivereal numbers such that limh→∞ rh = r. Thus, we can write

U(x, r) = ∪∞h=1B(x, rh) ,

and, by (3.17), we get the desired conclusion. Since by Lemma 1.16, A = ∪∞i=1Uiwith Ui (i ∈ N) open balls of Rn, by (3.17) we get the desired conclusion.

92

Proof of Proposition 3.28. Let us first observe that, by (3.24) and (3.25), it followsthat there exists c(s, n) := Hs(B(1)) ∈ [0,∞] such that (3.36) holds. We have onlyto show that

(3.37) 0 < c(n, n) = Hn(B(0, 1)) < ∞ .

Indeed, from Theorem 3.9 and (3.37), it will follow that c(s, n) = ∞, if s < n andc(s, n) = 0 if s > n. Let us observe that, by (3.10), the proof of left-hand sideinequality in (3.37) is equivalent to show that

(3.38) Sn(B(1)) > 0 ,

where Sn denotes the n-dimensional spherical Hausdorff measure on Rn. Let us

prelinimarly observe that, if B = B(x, r) is a closed ball of Rn and Q(i)B , Q

(c)B denote,

respectively, the inscribed and circumscribed n-dimensional (closed) cube to B, then

Q(i)B ⊂ B ⊂ Q

(c)B ,

and their side length are

l(Q

(i)B

)=

2r√n

and l(Q

(c)B

)= 2r .

In particular the diameters of Q(i)B and Q

(c)B are

(3.39) d(Q

(i)B

)= 2r = d(B) and d

(Q

(c)B

)=√n 2r =

√n d(B) .

Let us begin to prove (3.38). By definition of the n-dimensional pre-measure sphericalHausdorff measure and (3.39), we get, for each δ ∈ (0,∞],

Snδ (B(1)) = inf

αn

∞∑j=1

d(Bj)n : B(1) ⊂ ∪∞j=1Bj, Bj closed ball, d(Bj) 6 δ

= inf

αnnn/2

∞∑j=1

d(Q

(c)Bj

)n: B(1) ⊂ ∪∞j=1Bj, Bj closed ball, d(Bj) 6 δ

> inf

αn

∞∑j=1

(d (Qj)√

n

)n: B(1) ⊂ ∪∞j=1Qj, Qj closed cube

= inf

αn

∞∑j=1

Ln(Qj) : B(1) ⊂ ∪∞j=1Qj, Qj closed cube

> αn Ln(B(1)) > 0 ,

(3.40)

where the last inequality follows according to one of possible definitions of n- dimen-sional Lebesgue measure (see, for instance, [GZ, Sect. 4.3]). Passing to the limit as

δ → 0 in (3.40), (3.38) follows. Let Q∗ := Q(c)B(1) = [−1, 1]n, then the right-hand side

inequality in (3.37) will follow if we prove that

(3.41) Hn(Q∗) <∞ .

93

For each h ∈ N, let us divide Q∗ in hn closed cubes Qk (k = 1, . . . , hn) with sidelength 2/h. Then, for each δ ∈ (0,∞), by choosing h be such that

d(Qk) =

√n

2h< δ ∀ k = 1, . . . , hn ,

we get

Hnδ (Q∗) 6 αn

hn∑k=1

d(Qk)n =

αn nn/2

2n

hn∑k=1

1

hn=

αn nn/2

2n∀ δ ∈ (0,∞) .

By passing to the limit as δ → 0 in the previous inequality, (3.41) follows.

By means of Proposition 3.28 and Theorem 2.15, we can infer the agreement, upto a constant, of outer measures Hn and Ln on Rn.

Theorem 3.30. Let c :=Hn(B(1))

α∗n. Then

(3.42) Hn(A) = cLn(A) ∀A ⊂ Rn .

Proof. Assume that it holds

(3.43) Hn(B) = cLn(B) ∀B ∈ B(Rn) .

Let us prove that (3.42) also holds. Indeed, since Hn and Ln are regular Borel outermeasures, for each A ⊂ Rn there exist Borel sets Bi (i = 1, 2) such that

(3.44) A ⊂ B1 and Hn(B1) = Hn(A), A ⊂ B2 and Ln(B2) = Ln(A) .

By (3.43) and (3.44), it follows that

Hn(A) = Hn(B1) = cLn(B1) > cLn(A) ,

cLn(A) = cLn(B2) = Hn(B2) > Hn(A) .

Thus (3.42) follows. Let us now prove (3.43) . Let µ := Ln, ν :=1

cHn and λ :=

1

2(µ+ ν). Then, by Proposition 3.28, µ, ν and λ are positive Radon measures on

measure space (Rn,B(Rn)) and

(3.45) µ(B(x, r)) = ν(B(x, r)) = λ(B(x, r)) ∀x ∈ Rn, r ∈ (0,∞) .

Moreover it is trivial that

(3.46) µ << λ and ν << λ .

Thus, by (2.19),

(3.47) µ(B) =

∫B

Dλµ dλ and ν(B) =

∫B

Dλν dλ ∀B ∈ B(Rn) .

On the other hand, since

Dλµ(x) = Dλν(x) = 1 ∀x ∈ Rn ,

(3.43) follows.

We are now going to characterize the constant c in Theorem 3.30. Indeed we willprove that c = 1.

94

Theorem 3.31 (Hn ≡ Ln). Ln(A) = Hnδ (A) = Hn(A) for each A ⊂ Rn, 0 < δ 6 ∞.

The proof of Theorem 3.31 is based on the isodiametric inequality, which, as saidbefore, is rather complicated. Indeed it follows by Steiner symmetrization, which wewill not introduce here. Isodiametric inequality states that Euclidean balls in Rn

are the sets of maximum n-dimensional Lebesgue measure among all sets of a givendiameter.

Theorem 3.32 (Isodiametric inequality).

Ln(A) 6 αn d(A)n for A ⊂ Rn .

Proof. See [EG, Theorem 1, Sect. 2.2].

Remark 3.33. Notice that the isodiametric inequality is not trivial: a set of a givendiameter is not necessarily contained in a ball of the same diameter.

Exercise: Let A be an equilater triangle of the plane R2, with side length l.Prove that d(A) = l and there is no closed ball with diameter l containing A.

Proof of Theorem 3.31. 1st step. Let us first observe that, by Corollary 3.29 (i) ,Hn is a Radon measure on Rn.

2nd step. Let us prove the inequality

(3.48) Hnδ (A) 6 Ln(A) ∀A ⊂ Rn, 0 < δ 6 ∞ .

Let V ⊂ Rn be an open set such that A ⊂ V an let

Gδ :=

B(x, r) : x ∈ A, r < δ

2, B(x, r) ⊂ V

.

By Vitali’s covering property for Radon measures (see Theorem 2.11) the exists adisjoint subfamily F ⊂ Gδ, at most countable, such that

Hn(A \ ∪F) = 0 .

By Lemma 3.8, it also follows that

Hnδ (A \ ∪F) = 0 .

Therofore, since Hnδ is an outer measure,

Hnδ (A) 6 Hn

δ (∪F) 6∑B∈F

Hnδ (B)

6 αn∑B∈F

d(B)n = Ln(∪F) 6 Ln(V ) .

Taking the infimum in the previous inequality, over all open sets V ⊃ A, we get(3.48).

3rd step. Let us prove the inequality

(3.49) Hnδ (A) > Ln(A) ∀A ⊂ Rn, 0 < δ 6 ∞ .

Let (Ai)i be a sequence of sets such that

(3.50) A ⊂ ∪∞i=1Ai, d(Ai) < δ ∀ i .

95

Then, by the isodiametric inequality,

Ln(A) 6∞∑i=1

Ln(Ai) 6 αn

∞∑i=1

d(Ai)n .

Taking the infimum in the previous inequality, over all sequences (Ai)i satisfying(3.50), we get (3.49).

Hausdorff measures and Cantor sets

We have now introduced measures for measuring the size of very general sets. Wegive a look at some examples with which Hausdorff measures are convenient anduseful. We begin with the most classical as the Cantor sets.

Let us first introduce the Cantor sets in R. Let 0 < λ < 1/2 and define by stepsthe following intervals.Step k = 1: denote I0,1 := [0, 1] and let us delete the middle open interval of length(1− 2λ) d(I0,1), that is interval (λ, 1− λ). This yields 2 closed intervals

I1,1 =: [0, λ] and I1,2 =: [1− λ, 1] .

Let us defineC1(λ) := ∪2

j=1I1,j .

We continue this process of selecting two subintervals of each already given interval.

Step k = 2: Let us delete from intervals I1,1 and I1,2 the open middle intervals oflength (1− 2λ) d(I1,1) = (1− 2λ) d(I1,2) = (1− 2λ)λ. This yields 22 closed intervalswhich can denote from left to right as

I2,1, I2,2, I2,3, I2,4 .

Let us defineC2(λ) := ∪4

j=1I2,j .

Step k: if we have defined the 2k−1 intervals

Ik−1,1, . . . , Ik−1,2k−1 ,

we define 2k intervalsIk,1, . . . , Ik,2k

by deleting from the middle of each Ik−1,j an interval of length (1 − 2λ)d(Ik−1,j) =(1− 2λ)λk−1. All the intervals Ik,j thus obtained have length

d(Ik,j) = λk ∀ j = 1, . . . , 2k .

Let us defineCk(λ) := ∪2k

j=1Ik,j .

We define the limit set of this construction by

(3.51) C(λ) := ∩∞k=1Ck(λ) .

Then it is well-known that :

• C(λ) is an uncountable compact set ,

•C(λ) = ∅,• H1(C(λ)) = L1(C(λ)) = 0.

96

If λ = 1/3, C(1/3) is the celebrated Cantor middle-third set.We shall now study the Hausdorff measures and dimension of C(λ). As usual, it

is much simpler to find upper bounds than lower bounds for the Hausdorff measures.This is due to the definition: a suitable chosen covering will give an upper estimate,but a lower estimate requires finding an infimum over arbitrary coverings. By usingRemark 3.12, our goal is to find out a value s ∈ (0, 1) such that

(3.52) 0 < Hs(C(λ)) <∞

from which it will follow that α = Hdim(C(λ)).

Question: How can we guess a value s satisfying (3.52)?

A possible way is by using the self-similar structure of C(λ), of which we speakabout.

Let Si : R→ R (i = 1, 2)

S1(x) := λx, S2(x) := λx+ 1− λ .

Then Si are two similarities of R and

Exercise:

S1 (C(λ)) = C(λ) ∩ [0, λ], S2 (C(λ)) = C(λ) ∩ [1− λ, 1] .

(Hint: Let us first prove that C(λ) = S1(C(λ))∪S2(C(λ)), from which the desiredconclusion will follow.)

Therefore, by the previous exercise, (3.24) and (3.25), it follows that

Hs (C(λ)) = Hs (C(λ) ∩ [0, λ]) +Hs (C(λ) ∩ [1− λ, 1])

= Hs (S1(C(λ))) + Hs (S2(C(λ)))

= λsHs (C(λ)) + λsHs (C(λ))

= 2λsHs (C(λ)) .

Thus, if (3.52) holds, then 2λs = 1, or, equivalently,

(3.53) s =log 2

log 1λ

.

Theorem 3.34 (Hausdorff dimension of the Cantor sets in R). Let s be the value in(3.53) and let αs be the constant in (3.23). Then

(i) Hs (C(λ)) 6 αs <∞;(ii) Hs (C(λ)) > αs > 0,

In particular Hs (C(λ)) = αs and

Hdim(C(λ)) =log 2

log 1λ

.

Remark 3.35. Note that Hdim(C(λ)) measures the sizes of the Cantor sets C(λ) ina natural way: when λ increases, the sizes of the deleted holes decrease and the setsC(λ) become larger, and also Hdim(C(λ)) increases. Notice also that when λ runsfrom 0 to 1/2, Hdim(C(λ)) takes all the values between 0 and 1.

97

Proof. (i) Notice that, by definition,

C(λ) ⊂ Ck(λ) := ∪2k

j=1Ik,j∀ k

and so, if δ = λk

Hsλk(C(λ)) 6 αs

2k∑j=1

d(Ik,j)s = αs 2k λks = αs (2λs)k = αs .

Letting k →∞ in the previous inequality, we get the desired inequality.(ii) This lower bound is harder and we recommend [Ma, 4.10].

Several different Cantor-type sets can be constructed in Rn still for n = 1 that inhigher dimensions n > 2 (see [Ma, 4.11-13] and [Fa] for a deeper analysis).

Hausdorff measures as Radon measures

It is immediate that Hk E induces a positive finite measure in Rn whenever E isHk measurable andHk(E) < ∞. Conversely, in many applications one needs to knowwhether a given measure µ is representable in terms of the Hausdorff measure, or atleast needs to estimate the Hausdorff dimension of the set where µ is concentrated. Inorder to compare µ with Hk the natural idea is to look at the ratio µ(B(x, r))/α∗k r

k,and this motivates the following definition.

Definition 3.36 (k-dimensional densities). Let µ be a positive Radon measure in anopen set Ω ⊂ Rn and k > 0. The upper and lower k -dimensional densities of µ at xare respectively defined by

Θ∗k(µ, x) := lim supr→0+

µ(B(x, r))

α∗k rk

, Θ∗k(µ, x) := lim infr→0+

µ(B(x, r))

α∗k rk

If Θ∗k(µ, x) = Θ∗k(µ, x) their common value is denoted by Θk(µ, x) and this nota-tion is also used for Rm -valued Radon measures µ whenever the densities of theircomponent µi are defined, i.e. the i-th component of Θk(µ, x) is Θk(µi, x) for anyi = 1, . . . ,m. For any Borel set E ⊂ Rn we define also

Θ∗k(E, x) := lim supr→0+

Hk(E ∩B(x, r))

α∗k rk

, Θ∗k(E, x) := lim infr→0+

Hk(E ∩B(x, r))

α∗k rk

and, if they agree, we denote the common value of these densities by Θk(E, x).

Clearly Θ∗k(E, x) = Θ∗k(Hk E, x) and Θ∗k(E, x) = Θ∗k(Hk E, x). Using the leftcontinuity of (0,∞) 3 r 7→ µ(B(x, r)) it can be easily checked that all the densitiesare Borel functions of x. Now we see how the upper density Θ∗k(µ, x) can be used toestimate from below and from above µ with Hk, which will turn out to be very usefulin the topic of rectifiable sets.

Theorem 3.37 (Estimates of the upper density of a Radon measure). Let Ω ⊂ Rn

be an open set and µ a positive Radon measure in Ω. Then, for any t ∈ (0,∞) andany Borel set B ⊂ Ω the following implications hold:

(3.54) Θ∗k(µ, x) > t ∀x ∈ B ⇒ µ > tHk B ,

(3.55) Θ∗k(µ, x) 6 t ∀x ∈ B ⇒ µ 6 2k tHk B .

98

Proof. See, for instance, [AFP, Theorem 2.56].

Two fundamental consequences of Theorem 3.37, very useful in the study of measure-theoretic property of sets, are the following.

Corollary 3.38. Let k ∈ [0, n] and assume that E ⊂ Rn is Hk-measurable andHk(E) < ∞. Then

(3.56) ∃Θk(E, x) = limr→0+

Hk(E ∩B(x, r))

α∗k rk

= 0 for Hk-a.e. x ∈ Rn \ E ;

(3.57) 2−k 6 lim supr→0+

Hk(E ∩B(x, r))

α∗k rk

6 1 for Hk-a.e. x ∈ E .

Remark 3.39. If k = n, let us recall that, by the result concerning the density of aset (see Corollary 2.18)

limr→0+

Hn(E ∩B(x, r))

α∗n rn

= limr→0+

Ln(E ∩B(x, r))

Ln(B(x, r)= χE(x) Hn-a.e. x ∈ Rn .

It is not the case when k ∈ (0, n), even though k is integer. Indeed it is possible tohave

lim supr→0+

Hk(E ∩B(x, r))

α∗k rk

< 1

and

lim infr→0+

Hk(E ∩B(x, r))

α∗k rk

= 0

for Hk-a.e. x ∈ E even if 0 < Hk(E) < ∞, for a E not regular in measure-theoreticsense, that is if E is unrerctifiable (see Example 4.28).

Proof of Corollary 3.38. (i) Let t > 0 and let

Bt := x ∈ Rn \ E : Θ∗k(E, x) > t .Then, by (3.54) with µ = Hk E and B = Bt,

Hk(Bt) = (Hk Bt)(Bt) 61

tµ(Bt)

61

t(Hk E)(Rn \ E) = 0 ∀ t > 0 .

Thus (3.56) follows.(ii) Let us first prove the left inequality in (3.57), which amounts to prove that

(3.58) Hk(B) = 0 if B :=x ∈ E : Θ∗k(E, x) < 2−k

.

Let tj := 2−k(1− 1/j), if j > 2, and

Bj := x ∈ E : Θ∗k(E, x) 6 tj .Then, by (3.55) with µ = Hk E and B = Bj,

Hk(Bj) = (Hk Bj)(Bj) 6 2ktj µ(Bj)

= 2ktjHk(Bj) ∀ j > 2 .

99

Since 2ktj < 1 for each j > 2, Hk(Bj) = 0 for each j, which implies

Hk(B) = Hk(∪∞j=2Bj) 6∞∑j=2

Hk(Bj) = 0

and then (3.58) follows. Finally the proof of the right inequality in (3.57) is similar.Indeed this amounts to prove that

(3.59) Hk(B) = 0 if B := x ∈ E : Θ∗k(E, x) > 1 .

Let tj := 1 + 1/j and

Bj := x ∈ E : Θ∗k(E, x) > tj .Then, applying (3.54) , we get that Hk(Bj) = 0 for each j, which yields the desiredconclusion.

3.4. Area and coarea formulas in Rn and some applications.

Some recalls of linear algebra

Before stating the two main results (Theorems 3.48 and 3.11) of this subsection,we recall some notations and results of linear algebra.

Notation: If L : Rn → Rm is a linaar map, we identify L with its transformationmatrixML, that is the (m×n)- matrix which represents L with respect to the standardbases of Rn and Rm. Moreover, if n = m we define detL := detML.

Definition 3.40. (i) Given L : Rn → Rm linear map, its norm ‖L‖ is defined by

(3.60) ‖L‖ := sup |L(v)| : |v| 6 1

(ii) A linear map O : Rn → Rm is orthogonal if

(Ox,Oy)Rm = (x, y)Rn ∀x, y ∈ Rn.

(iii) A linear map S : Rn → Rn is symmetric if

(Sx, y)Rn = (x, Sy)Rn ∀x, y ∈ Rn.

(iv) Let L : Rn → Rm be a linear map. The adjoint of L is the linear mapLT : Rm → Rn defined by

(x, LTy)Rn = (Lx, y)Rm ∀x ∈ Rn, y ∈ Rm.

(iv) Let L : Rn → Rm be a linear map. Then the rank of L, denoted rank(L) isthe rank of its associated transformation matrix ML, that is, the maximunnumber of columns ( or rows) of ML linearly independent.

Remark 3.41. (i) If L : Rn → Rm is a linear map, the, L is Lipschitz functionand ‖L‖ = Lip(L).

(ii) If O : Rn → Rm is an orthogonal map, then it is injective and so n 6 m.

Theorem 3.42 (Polar Decomposition). Let L : Rn → Rm be a linear function. Then

100

(i) If n 6 m there are a symmetric linear function S : Rn → Rn and an orthog-onal linear function O : Rn → Rm such that

L = O S.Moreover it holds

|detS| =√

det(LT L) .

(ii) If n > m there are a symmetric linear function S : Rm → Rm and anorthogonal linear function O : Rm → Rn such that

L = S OT .

Moreover it holds|detS| =

√det(L LT ) .

Proof. See [EG, Sect. 3.2].

Definition 3.43 (Jacobian of a linear map). Assume L : Rn → Rm be a linearfunction.

(i) If n 6 m and assume that L = O S as above. We define the Jacobian of Lto be

JL := |detS| =√

det(LT L) .

(ii) If n > m and assume that L = S OT as above. We define the Jacobian of Lto be

JL := |detS| =√

det(L LT ) .

Remark 3.44. (i) It follows from Theorem 3.42 that the definition of JL is in-dependent of the particular choice of S and O.

(ii) Clearly JL = JLT .

Theorem 3.45 (Binet-Cauchy formula). Assume that n 6 m and let L : Rn → Rm

be a linear map. Then

(3.61) JL =

√ ∑N⊂ML

(detN)2

where the sum is understood over each (n×n)-submatrix N of (m×n)- transformationmatrix ML of L.

Proof. See [EG, Sect. 3.2].

Remark 3.46. From the definition of jacobian and Binet-Cauchy formula (3.61), wecan infer that, if L : Rn → Rm is a linear map, then

JL = 0 if and only if rank(L) < minn,m .

Area formula

We are going to prove that the integral n-dimensional Hausdorff measure Hn withn = 1, 2, . . . ,m turns out to be the classical surface n-measure ( or, also, n-volume)for regular n-dimensional submanifold of Rm. Let us recall that, by Theorems 3.25and 3.31, we partially met this goal for cases k = 1, m. We are going to accomplishthe task in the remaining cases by means of the area formula.

101

A possible way for introducing a regular n-dimensional submanifold of Rm (n 6 m)is through a (regular) parametrization (see, for instance, [Fe, 3.1.19]). For instance,a n- regular submanifold Γ ⊂ Rm can be given as follows . Assume that

Γ = f(A)

where A ⊂ Rn is a regular bounded subset and f : A ⊂ Rn → Rm, called parametriza-tion of Γ, satisfies

• f : A→ Rm is injective;• f ∈ C1(A;Rm) and df : A→ Rm has maximal rank.

If this is the case, it well-known that the n- volume of Γ, is defined as

n− volume(Γ) :=

∫A

Jf(x) dLn(x) ,

where Jf(x) denotes the Jacobian of f at x. Jacobian Jf is the corrective factorrelating the elements of volumes of the domain and image of f .Example: (Jacobian for a surface in R3) Suppose that n = 2 and m = 3, Γ =f(A), with (s, t) ∈ A ⊂ R2 and f = f(s, t) = (f1(s, t), f2, (s, t), f3(s, t)) : A ⊂R2 → R3 satisfying the previous assumptions. Let ∂sf := (∂sf1, ∂sf2, ∂sf3), ∂tf :=(∂tf1, ∂tf2, ∂tf3) and let Ni (i = 1, 2, 3) denote (all) the 2× 2-submatrices of the 3× 2Jacobian matrix of f

Df(s, t) :=

∂sf1(s, t) ∂tf1(s, t)∂sf2(s, t) ∂tf2(s, t)∂sf3(s, t) ∂tf3(s, t)

Then it is well-known by differential geometry that the Jacobian of f at (u, v)

equals

Jf(s, t) = |∂sf ∧ ∂tf |(s, t) =√

(detN1)2 + (detN2)2 + (detN3)2(s, t) (s, t) ∈ A ,

where v ∧ w denotes the exterior product of vectors v, w ∈ R3.

Problem: Hn(Γ) = n− volume(Γ)?A positive answer is a particular case of the area formula.

Definition 3.47. Assume f : Rn → Rm be a differentiable at x ∈ Rn.

(i) If n 6 m, we define the Jacobian of f at x to be

Jf(x) := Jdf(x) =√

det (Df(x)T ·Df(x)) ;

(ii) If n > m, we define the Jacobian of f at x to be

Jf(x) := Jdf(x) =√

det (Df(x) ·Df(x)T ) ;

where Df(x) denote the m× n-Jacobian matrix of f at x.

According to the previous definition, if f : Rn → Rm is Lipschitz with n 6 m, wedefine as Jacobian of f the Borel measurable function Jf : Rn → [0,∞] defined as(3.62)

Jf(x) :=

√det (Df(x)T ·Df(x)) if f is differentiable at x

∞ if f is not differentiable at xif n 6 m,

102

and(3.63)

Jf(x) :=

√det (Df(x) ·Df(x)T ) if f is differentiable at x

∞ if f is not differentiable at xif n > m.

Notice that, by Rademacher’s theorem, set Jf < ∞ coincides with the set of pointsx ∈ Rn at which f is differentiable and it has full Lebesgue measure in Rn.

Theorem 3.48 (Area formula). Let f : Rn → Rm be Lipschitz with n 6 m. Thenfor each Ln-measurable subset A ⊂ Rn

(AF)

∫A

Jf dLn =

∫RmH0(A ∩ f−1(y)

)dHn(y) .

Function Rm 3 y 7→ H0 (A ∩ f−1(y)) ∈ N ∪ ∞ is called multiplicity function off .

Remark 3.49. (i) Notice that, if y /∈ f(A), then H0 (A ∩ f−1(y)) = 0. Hence(AF) can be equivalently written

(3.64)

∫A

Jf dLn =

∫f(A)

H0(A ∩ f−1(y)

)dHn(y) .

(ii) It follows that, if n 6 m, f : Rn → Rm is Lipschitz and A is bounded, then,by (AF),

H0(A ∩ f−1(y)

)< ∞ Hn-a.e. y ∈ Rm .

Therefore f−1(y) is at most countable for Hn-a.e. y ∈ Rm.

Area formula (AF) immediately yields a positive answer to the previous question.

Theorem 3.50 (Area formula for injective maps). Let n 6 m and let f : Rn → Rm

be an injective Lipschitz function and A ⊂ Rn be a measurable set. Then

(IAF) Hn(f(A)) =

∫A

Jf dLn

and Hn f(Rn) is a Radon measure on Rm.

We are now going to give an idea of the proof of the area formula for injective maps,that is Theorem 3.50. Proof of the general area formula, that is Theorem 3.48, canbe obtained by using the are formula formula for injective maps (see [Mag, Theorem8.9]).

We preliminarly need the following fundamental results.

Lemma 3.51 (Measurability of Lipschitz functions images ). If n 6 m, E is aLebesgue measurable set in Rn and f : Rn → Rm is a Lipschitz function, then f(E)is Hn-measurable in Rm.

Proof. See [EG, Lemma 2, Sect. 3.3.1] or [Mag, Lemma 8.4].

Lemma 3.52 (Area formula for linear functions). Let n 6 m and let L : Rn → Rm

be a linear function. Then for all A ⊂ Rn

(3.65) Hn(L(A)) = JLLn(A)

103

Remark 3.53. Notice that that an alternative definition of Jacobian JL for a linearmap L : Rn → Rm (n 6 m) can be as value Hn(L(Q)) where Q = [0, 1]n is the unitcube of Rn.

Proof. See [EG, Lemma 1, Sect. 3.3.1] or [Mag, Theorem 8.5].

Lemma 3.54 (Role of the singular set of a Lipschitz map). Let n 6 m and f : Rn →Rm be Lipschitz. Then

Hn (f(x ∈ Rn : Jf(x) = 0)) = 0 .

Proof. See [Mag, Theorem 8.7].

Lemma 3.55 (Lipschitz linearization). Let n 6 m and f : Rn → Rm be Lipschitz.Let us fix (an arbitrary) t > 1. Denote

F := x ∈ Rn : 0 < Jf(x) < ∞ ,then there exists a countable disjoint family of Borel sets (Fh)h ∈ N such that

(i) F = ∪∞h=1Fh;(ii) f |Fh is injective;(iii) for each h, there exists a symmetric automorphism Sh : Rn → Rn such that

f |Fh S−1h : Sh(Fh) ⊂ Rn → f(Fh) ⊂ Rm is a bi-Lipschitz map and the

following estimates hold: for every x, y ∈ Fh and v ∈ Rn

(3.66) Lip(f |Fh S−1

h

)6 t, Lip

(Sh (f |Fh)−1

)6 t

(3.67)1

t|Shv| 6 |Df(x)v| 6 t |Shv|,

(3.68)1

tnJSh 6 Jf(x) 6 tn JSh.

Proof. The primitive proof is in [Fe, 3.2.2]; see also [EG, Lemma 3, Sect. 3.3.1] and[Mag, Theorem 8.8].

Proof of Theorem 3.50.Step 1: Let us prove that

(3.69) Hn(f(A)) = Hn(f(A ∩ F )) ,

where F is the set in Lemma 3.55. Let us recall that, by Theorem 3.22 and theagreement Ln = Hn on Rn (see Theorem 3.31)

(3.70) Hn(f(E)) 6 Lip(f)n Ln(E) ∀E ⊂ Rn .

Thus both sides of (IAF) are zero whenever Ln(A) = 0. By Rademacher’s theoremand Lemma 3.54, we get

Hn(f(A ∩ F )) 6 Hn(f(A)) = Hn(f((A ∩ F ) ∪ (f(A \ F )))

6 Hn(f(A ∩ F ) +Hn(f(A \ F ))

6 Hn(f(A ∩ F ) +Hn(f(Jf = 0)) +Hn(f(Jf =∞))= Hn(f(A ∩ F )) ,

which shows (3.69).

104

Step 2: Let us prove (IAF). By the previous step, we can assume that A ⊂ F .We now fix t > 1 and consider the partition (Fh)h∈N of F given by Lemma 3.55. Wesee A as the union of the disjoint sets (Fh ∩ A)h, so that, by the global injectivityof f , we have that f(A) is the disjoint union of the sets (f(Fh ∩ A))h, which areHn-measurable by Lemma 3.51. Therefore, by Theorem 3.22, the linear case of thearea formula (3.65) and (3.66), we find that

Hn(f(A)) =∞∑h=1

Hn (f(A ∩ Fh)) =∞∑h=1

Hn((f |Fh S−1

h ) (Sh(A ∩ Fh)))

6∞∑h=1

Lip(f |Fh S−1h )nHn (Sh(A ∩ Fh))

6 tn∞∑h=1

JSh Ln(A ∩ Fh)

6 t2n∞∑h=1

∫A∩Fh

Jf(x) dx = t2n∫A

Jf(x) dx ∀ t > 1 .

(3.71)

In a similar way, by analogous argument∫A

Jf(x) dx =∞∑h=1

∫A∩Fh

Jf(x) dx 6 tn∞∑h=1

JSh Ln(A ∩ Fh)

= tn∞∑h=1

Hn((Sh (f |Fh)−1)f(A ∩ Fh)

)6 t2n

∞∑h=1

Hn (f(A ∩ Fh)) = t2nHn (f(A)) ∀ t > 1 .

(3.72)

We thus prove (IAF) by letting t→ 1+ in (3.71) and (3.72).Step 3: Let us prove thatHn f(Rn) is a Radon measure. By Lemma 3.51, f(Rn)

is Hn-measurable, while (IAF) implies Hn f(Rn) to be locally finite. By Theorem1.94, Hn f(Rn) is a Radon measure.

Some applications of the area formula:

(1) (Length of a curve). Assume that n = 1, m > 1 and f : R → Rm is aninjective Lipschitz function. In this case, the Jacobian matrix of f

Df(t) = f ′(t)T =

f ′1(t). . .. . .f ′m(t)

m×1

L1-a.e. t ∈ R ,

and according to definition of the Jacobian of f

Jf(t) =√

det (Df(t)T ·Df(t)) = |f ′(t)| L1−a.e.t ∈ R .

105

Thus, by (IAF), for a, b ∈ R with a < b,

H1(Γ) =

∫ b

a

|f ′(t)| dt if Γ := f([a, b]) .

(2) (Area of a graph). Assume that n > 1, m = n + 1 and u : Rn → R is aLipschitz function. Let f : Rn → Rn+1 defined as

f(x) := (x, u(x)) x ∈ Rn .

Then

Df(x) =

1 0 . . . . . . 00 1 0 . . . 0. . . . . . . . . . . . . . .0 0 . . . 0 1

∂1u(x) . . . . . . . . . ∂nu(x)

(n+1)×n

Ln-a.e. x ∈ Rn ,

and, by applying Binet-Cauchy formula (3.61),

(3.73) Jf(x) =√

det (Df(x)T ·Df(x)) =√

1 + |∇u(x)|2 Ln-a.e. x ∈ Rn .

Thus, if A ⊂ Rn is an open set and Γ = f(A) = graph(u;A) := (x, u(x)) :x ∈ A,

(3.74) Hn(Γ) =

∫A

√1 + |∇u(x)|2 dLn(x) .

Finally let us state a general formula for the change of variables.

Theorem 3.56 (Change of variables). Let n 6 m and f : Rn → Rm be Lipschitz.The for each Ln-integrable function g : Rn → R,

(3.75)

∫Rng(x) Jf(x) dLn(x) =

∫Rm

∑x∈f−1(y)

g(x)

dHn(y) .

In particular, if f is injective,

(3.76)

∫Rng(x) Jf(x) dLn(x) =

∫f(Rn)

g(f−1(y)) dHn(y) .

Proof. See [EG, Theorem 2, Sect. 3.3.3].

Remark 3.57. Recall that f−1(y) is at most countable Hn-a.e. y ∈ Rm (see Remark3.49 (ii)).

Coarea formula

We are now going to presente a far-reaching generalization of Fubini’s theorem (seeTheorem 1.101), which is very useful in GMT and, more generally, in analysis. Letus begin with an example.

Example: Let n = 2 and Q = [0, 1]2 and let L : R2 → R be the linear mapL(x1, x2) := x1, that is the (orthogonal) projection on the x1-axis. If t ∈ R, then

L−1(t) = (t, x2) : x2 ∈ R ,

106

that is, L−1(t) is a straight line parallel to the x2-axis. It is trivial to see that, if wedenote At := s ∈ R : (t, s) ∈ A, then

L1(At) = H1(Q ∩ L−1(t)) ∀ t ∈ R ,and, by Fubini’s theorem, we can write that, if n = 2, m = 1,

H2(Q) = L2(Q) =

∫RL1(At) dt =

∫RHn−m(Q ∩ L−1(t)) dt .(3.77)

Let us now consider the linear map L(x1, x2) = −x1 + x2. If t ∈ R, then

L−1(t) = (x1, x1 + t) : x1 ∈ R ,that is L−1(t) is still a straight line, but it is not parallel to the coordinate axes. In

particular identity (3.77) no more holds. Indeed, we can get, by a simple calculation,that

H1(Q ∩ L−1(t)) =

√2(1− |t|) if |t| 6 1

0 if |t| > 1

and ∫RH1(Q ∩ L−1(t)) dt =

√2 =√

2H2(Q) .

Problem: Let n > m and let L : Rn → Rm be a linear function, find out a nonnegative factor c(L) such that

(3.78)

∫RmHn−m(A ∩ L−1(y)) dLm(y) = c(L)Ln(A) ∀A ⊂ Rn .

Theorem 3.58 (Coarea for linear maps). Let n > m and let L : Rn → Rm be linear.Then (3.78) holds with

c(L) := JL =√

det(L LT )

where LT : Rm → Rn denotes the adjoint linear map of L.

Proof. See [EG, Lemma 1, Sect. 3.4.1].

Theorem 3.59 (Corea formula). Let n > m and let f : Rn → Rm be a Lipschitzfunction. The for each Ln-measurable set A ⊂ Rn∫

A

Jf(x) dLn(x) =

∫RmHn−m(A ∩ f−1(y)) dLm(y) ,

where Jf is the Jacobian factor defined in (3.63).


Remark 3.60. Applying the coarea formula to set A := x ∈ Rn : Jf(x) = 0, weget that

(WMS) Hn−m (Jf = 0 ∩ f−1(y))

= 0 Lm-a.e. y ∈ Rm .

This is a weak variant of Morse-Sard’s theorem which asserts

(MS) Jf = 0 ∩ f−1(y) = ∅ Lm-a.e. y ∈ Rm ,

provided that f ∈ Ck(Rn;Rm) for k = 1 + n −m. Observe, however, that (WMS)only requires that f be Lipschitz.

107

Theorem 3.61 (Change of variables formula). Under the same assumptions of theCoarea Formula. Then for each Ln-measurable function g : Rn → R

(i) g|f−1(y) is Hn−m-summable Lm-a.e. y ∈ Rm.(ii) ∫ n

Rg(x) Jf(x) dLn(x) =

∫Rm

[∫f−1(y)

g(x) dHn−m(x)

]dLm(y) ,



About Hausdorff measures and length measure

Let (X, d) be a metric space. A curve of X is a continuous function γ : [a, b]→ X;its support is the subset Γ = γ([a, b]) ⊂ X and, in this case, γ is called a parametriza-tion of Γ. Given a curve γ : [a, b] → X and a subinterval [c, d] ⊆ [a, b], we definevariation of γ on [c, d]

Var(γ; [c, d)) := sup

m∑i=1

d(γ(ti), γ(ti−1)) : t0 = c < t1 < · · · < tm = d

∈ [0,∞] .

By analogy with the case where X = Rn and d is the Euclidean distance, the quantityVar(γ; [c, d)) represents the length (with respect to the metric d) of curve γ over [c, d].

Therehore, given a subinterval [c, d] ⊆ [a, b], we define the length of γ over [c, d] as

(3.79) l(γ; [c, d]) := Var(γ; [c, d)) .

Thus we can define as length of γ the quantity

l(γ) := l(γ; [a, b]) .

We say that γ is rectifiable if l(γ) < ∞. It is easy to see that, if γ : [a, b] → X is acurve, then

(3.80) l(γ; [c, d]) > d(γ(c), γ(d)),whenever a 6 c 6 d 6 a ;

and

(3.81) l(γ; [a, b]) = l(γ; [a, c]) + l(γ; [c, b]), whenever a 6 c 6 b .

Theorem 3.62 (Classical length and H1). Let γ : [a, b]→ X be a curve and denoteΓ = γ([a, b]) its support. Then

H1(Γ) 6 S1(Γ) 6 l(γ)

and equality holds if γ is injective, where S1 denotes the 1-dimensional sphericalHausdorff measure defined in Remark 3.5.

Proof. Proof’s strategy is similar to the one of Theorem 3.25 by means of suitablechanges (see [AT, Theorem 4.4.2] and [SC, Theorem 2.29]).

108

About Hausdorff dimension in a metric measure space

An useful criterion to estimate the Hausdorff dimension of a metric measure spaceis the following.

Theorem 3.63. Let (X, d, µ) be a metric measure space with (X, d) separable. Sup-pose that there exist constants c > 1 and Q > 0 such that

(3.82)1

crQ 6 µ(B(x, r)) 6 c rQ ∀ 0 < r < d(X) .

Then there exists a constant c′ > 1 such that

(3.83)1

c′HQ(E) 6 µ(E) 6 c′HQ(E) ,

for each Borel set E ⊂ X. In particular

Hdim(X) = Q .

Proof. See, for instance, [SC, Theorem 2.26].

Metric measure spaces where formula (3.82) holds are called Ahlfors regular ofdimension Q. By (3.83), we can replace µ by the Hausdorff Q-measure in an Q-regular space without essential loss of information.

About Lipschitz maps

Extension’s problem for Lipschitz maps acting between metric spaces is an issue ofthe current research in analysis in metric spaces (see [AT, Chap. 3], [He, Chap. 6]and [He2]). The proof of Kirszbraun’s theorem depends crucially on very euclideanproperties of the domain space Rn. Hence its proof cannot be extended to generalmetric spaces.

Differentiability for Lipschitz functions f : Rk → (X, d) has been studied. Inparticular Rademacher-type theorems have been obtained (see [AK]).

A very important Rademacher type-result for real valued Lipschitz functions de-fined on a metric measure space (X, d, µ), that is Lipschitz functions f : (X, d, µ)→R, has been obtained by Cheeger [Ch] and it has been a seminal paper for the de-velopement of analysis in metric measure spaces (see, for instance, [AG] and thereferences therein).

About Lipschitz maps and Hausdorff measures

Notice that the conclusion of Theorem 3.22 holds for each Lipschitz map f :(X, d)→ (Y, %).

Notice also that for every isometric embedding f : (X, d) → (Y, %), that is amapping satisfying %(f(x), f(y)) = d(x, y), by definition of Hausdorff measure, itfollows that

Hs(f(A)) = Hs(A) ∀A ⊂ X .

About area and coarea formulas

109

Area and coarea formulas have been obtained respectively for Lipschitz functionsf : Rk → (X, d) and f : (X, d)→ Rk (see [AK]).

4. Rectifiable sets and blow-ups of Radon measures ([AFP, Mag, Ma])

Motivation: the introduction of the notion of rectifiable set in Rn, which is a setsmooth in a certain measure-theoretic sense. It is an extension of the idea of arectifiable curve to higher dimensions and it has many of the desirable propertiesof smooth manifolds, including tangent spaces that are defined almost everywhere.Rectifiable sets are fundamental objects of study in geometric measure theory.

k-dimensional planes and the orthogonal group in Rn.

Let us begin to recall the notion of k-dimensional plane in Rn and some theirproperties.

We simply mean by a k-dimensional plane π in Rn a k-dimensional subspace ofRn. We will denote by

G(n, k)

the class of k-dimensional planes of Rn, which is also called Grassmannian manifoldof k dimensional subspaces of Rn.

If π is a k-dimensional plane we denote by π⊥ the (n − k)-plane orthogonal to π,that is

π⊥ := x ∈ Rn : (x, y)Rn = 0 ∀ y ∈ π .Notice also that, if π ⊂ Rn is a k-plane and v1, . . . , vk is an orthonormal basis of πthen the map Iπ : Rk → Rn

(4.1) Iπ(x) :=k∑j=1

xj vj if x ∈ Rk

defines an (injective) orthogonal map such that Iπ(Rk) = π.Given a k-dimensional plane π in Rn, we denote by Pπ : Rn → Rn and P⊥π : Rn →

Rn respectively the orthogonal projections of Rn onto π and π⊥ . In particular itturns out that P⊥π = Pπ⊥ .

Tipically we identify π with Pπ and we endow G(n, k) by the distance

(4.2) |π1 − π2|G := ‖Pπ1 − Pπ2‖ = supv∈Sn−1

|Pπ1(v)− Pπ2(v)| if πi ∈ G(n, k) i = 1, 2 ,

that is ‖ · ‖ denotes the norm in (3.60).


(4.3) (G(n, k), | · |G) is a compact metric space.

The orthogonal group O(n) of Rn consists of all linear orthogonal maps L : Rn →Rn according to Definition 3.40 (ii), that is linear maps preserving the inner product,

(L(x), L(y))Rn = (x, y)Rn ∀x, y ∈ Rn ,

110

or equivalently L is an isometry,

|L(x)− L(y)| = |x− y| ∀x, y ∈ Rn .

It is easy to see that O(n) turns out to be a group with composition as group law. Themembers of O(n) consist of rotations and rotations composed with a reflexion oversome hyperplane. Another way to view them is to observe that they map orthonormalbasis to orthonormal basis, and conversely given two orthonormal bases u1, . . . , un andv1, . . . , vn of Rn one can define L ∈ O(n) by setting L(ui) = vi and extending linearly.

One of the basic properties of O(n) is that it acts transitively on Sn−1: for anyx, y ∈ Sn−1 there exists L ∈ O(n) such that L(x) = y.

In the case n = 2, 0(2) is very simple. It consists of rotations around the originand of rotations composed with the reflexion over the x-axis.

By simple linear algebra the action of O(n) on G(n, k) is distance-preserving:

|L(π1)− L(π2)|G = |π1 − π2|G ∀L ∈ O(n), πi ∈ G(n,m) i = 1, 2 .

Also O(n) transitively acts on G(n, k): for πi ∈ G(n, k), i = 1, 2, there is L ∈ O(n)such that L(π1) = π2, that is π1 and π2 are isometric. To see this, take orthonormalbases for π1 and π2, complete them to orthonormal bases of Rn, and choose L ∈ O(n)which maps one of these onto the other.

In particular it follows, if B1 = B(1) denotes the unit closed ball in Rn centered at0 and π is a k-dimensional plane in Rn, then set π∩B1 is isometric to the unit closedball in Rk and so, by Remark 3.24,

(4.4) Hk(π ∩B1) = α∗k .

Analogously, set π ∩ ∂B1 is isometric to the unit sphere in Rk and so

(4.5) Hk(π ∩ ∂B1) = 0 ,

Regular surfaces and rectifiable sets.

Let us now recall the classical notion of regular surface in Rn.

Definition 4.1. Given k ∈ N, 1 6 k 6 n − 1, m > 1, we shall say that Γ ⊂ Rn

is a k-dimensional (embedded) surface (or submanifold) of class Cm in Rn (or a Cm-hypersurface when k = n − 1) if for every x ∈ Γ there exist an open neighborhoodV ⊂ Rn of x, an open set U ⊂ Rk and a bijection f : U → V ∩ Γ with f ∈ Cm(U)and df(x) : Rk → Rn is injective for each x ∈ U . Each map f is called a coordinatemapping of Γ.

Remark 4.2. (i) It is well-known from standard calculus that df(x) : Rk → Rn

is injective if and only if Jf(x) > 0, where Jf(x) denotes the Jacobian of fat x (see Definition 3.47 and Remark 3.46).

(ii) Notice that, in this way, Γ is relatively open in Rn, and can be covered bycountably many images f(U), with f and U as above.

The notion of rectifiable set is just a generalization of Remark 4.2 (ii) to themeasure-theoretic setting.

Definition 4.3. Let 1 6 k 6 n− 1 be integers and let Γ ⊂ Rn.

111

(i) We say that Γ is countably Hk-rectifiable if

(CR) Γ = Γ0 ∪ (∪∞i=1fi(Ai)) ,

whereHk(Γ0) = 0, Ai ⊂ Rk are Lk-measurable and fi : Ai → Rn are Lipschitzfunctions for i = 1, 2, . . . .

(ii) We say that Γ is locally Hk-rectifiable if Γ is countably Hk-rectifiable andHk(Γ ∩K) < ∞ for each compact K ⊂ Rn.

(iii) We say that Γ isHk-rectifiable if Γ is countablyHk-rectifiable andHk(Γ) < ∞.

Remark 4.4. Since Lipschitz functions defined on subsets of Rk can be extended toall of Rk, keeping a control of the Lipschitz constant (see Theorem 3.18), then we canequivalently say that Γ is countably k-rectifiable if it is Hk-measurable and

(CR*) Γ ⊂ Γ0 ∪ (∪∞i=1Γi) ,

where Hk(Γ0) = 0, Γi := fi(Rk) and fi : Rk → Rn are Lipschitz functions fori = 1, 2, . . . .

Remark 4.5. Observe that the rectifiability is a metric notion since it depends onthe notion of Lipschitz maps acting between metric spaces (see section IV.4).

Historical notes: The concepts of rectifiability were first introduced for one-dimensionalsets in R2 by Besicovitch [Be3] in 1928 and then subsequently studied in papers[Be4, Be5] for k = 1 and n = 2. Federer extended the study for general k and n in1947 and those results are contained in his celebrated monograph [Fe].

Example 4.6 (Lipschitz k-graph). Let π be a k-plane and let φ : π → π⊥ be aLipschitz function.We call graph of φ the set

Γ = graph(φ) := z + φ(z) : z ∈ π ⊂ Rn .

Let us define f : Rk → Rn

f(x) := Iπ(x) + φ(Iπ(x)) x ∈ Rk ,

where Iπ : Rk → Rn is the map defined in (4.1). It is easy to check that f is stillLipschitz and, from Theorem 3.22, we get that Γ is locallyHk-rectifiable. In particularany compact subset of Γ is Hk-rectifiable.

Remark 4.7. Observe that that countable Hk- rectifiability (CR*) is equivalent tothe seemingly stronger requirement that sets Γi (i = 1, 2, . . . ) is a Lipschitz k-graphs(see [AFP, Proposition 2.76]). Using Whitney’s extension theorem (see Theorem3.14) to approximate Lipschitz functions by C1 functions, it could be possible toshow that Γi (i = 1, 2, . . . ) can be a C1 k-graphs (see [Fe, 3.1.16]). However, sinceLipschitz functions are more flexible than C1 functions in many typical constructionsof geometric measure theory (see Mc Shane’s extension theorem 3.16), the use ofLipschitz functions is preferred.

In this chapter we are going to study some geometric properties of locally Hk-rectifiable sets, such as the existence Hk-a.e. of a tangent plane in a suitable senseand a their characterization in terms of measure-theoretic properties. We will strictlyfollow the arguments in [Mag, Chap. 10].

Observe that, whenever Γ is a countably Hk-rectifiable, then Hk Γ is a regularBorel outer measure. However, from Theorem 1.94, Hk Γ is a Radon measure if

112

and only if Γ is locally Hk-rectifiable. Therefore, it is under the assumption of localHk-rectifiability on Γ that we have a natural identification between Γ and a Radonmeasure µ. In turn, as seen in Example 1.111, this identification lies at the basis of themeasure-theoretic formulation of the notion of tangent space. Indeed, if Γ is locallyHk-rectifiable and µ = Hk Γ, then for Hk-a.e. x ∈ Γ there exists a k-dimensionalplane πx in Rn such that the blow-ups µx,r of µ at x weak* converge to Hk πx asr → 0+, that is

(4.6) Hk

(Γ− xr

)∗ Hk πx as r → 0+ .

A crucial fact is that the converse also holds true: if µ is a Radon measure onRn concentrated on a Borel set Γ and such that for every x ∈ Γ there exists a k-dimensional plane πx such that the k-dimensional blow- ups of µ have the propertythat

(4.7) µx,r =(Φx,r)#µ

rk∗ Hk πx, as r → 0+,

then Γ is locally Hk rectifiable and µ = Hk Γ where Φx,r : Rn → Rn is the mapdefined by

Φx,r(y) :=y − xr

y ∈ Rn .

4.1. Rectifiable sets of Rn and their decomposition in regular Lipschitzimages. In this section we are going to replace the decomposition of a countably Hk-rectifiable set, provided in (CR), with a ”good decompostiton” composed or ”regular”Lipschitz parametrization maps fi : Ai ⊂ Rk → Rn. Let us begin to introduce thenotion of regular parametrization Lipschitz map.

Definition 4.8 (Regular Lipschitz image). Given a Lipschitz map f : Rk → Rn anda compact set E ⊂ Rk, we say that the pair (f, E) defines a regular Lipschitz imagef(E) in Rn if

(i) f is injective and differentiable on E, with Jf(x) > 0 for each x ∈ E;(ii) Lk-a.e. x ∈ E is a point of density 1 for E;

(iii) Lk-a.e. x ∈ E is a Lebesgue point of Df .

Example 4.9. A k-regular C1 surface Γ ⊂ Rn can be seen as countable union ofregular Lipschitz images. Indeed, according to Definition 4.1 and Remark 4.2, Γcan be covered by a countable union of relatively compact open sets (Vi)i of Rn forwhich there exists a countable family of coordinate mappings fi : Ui → Γ ∩ Vi,which are bijective, with (Ui)i relatively compact open sets of Rk, fi ∈ C1(Ui;Rn),dfi(x) : Rk → Rn is injective for each x ∈ Ui and Jfi(x) > 0 for each x ∈ Ui.Moreover, since each map fi : Ui → Rn is a Lipschitz function, it can be extended, asa Lipschitz function, to the whole Rk (see Corollary 3.17) . Therefore the sequenceof couples (fi, Ei), with Ei := Ui yield a countable union of regular Lipschitz imagesf(Ei).

Remark 4.10. In particular, we immediately deduce from (ii) and (iii) of Definition4.8 that

limr→0+

|E ∩B(x, r)|α∗k r

k= 1, lim

r→0+

1

rk

∫B(x,r)

|Jf(x)− Jf(y)| dy = 0

113

for every x ∈ E. Indeed, if x is a Lebesgue point of the Jacobian matrix of f , Df ,then x is a Lebesgue point of Jf , since Df ∈ L∞(Rk;Mn,k) and the map L(Rk;Rn) 3L 7→ JL is continuous, where Mn,k and L(Rk;Rn) denote respectively the class of realmatrices with n rows and k columns and the space of linear maps from Rk to Rn .

We now show that we can always decompose a countably Hk- rectifiable set bymeans of (almost flat) regular Lipschitz images (see also Lemma 3.11).

Theorem 4.11 (Decomposition of rectifiable sets). If Γ is countably Hk- rectifiablein Rn and t > 1, then there exist a Borel set Γ0 ⊂ Rn, countably many Lipschitzmaps fh : Rk → Rn and compact sets Eh ⊂ Rk such that

(4.8) Γ = Γ0 ∪ (∪∞h=1fh(Eh)) , Hk(Γ0) = 0 .

Each pair (fh, Eh) defines a regular Lipschitz image, with Lip(fh) 6 t and

(4.9) t−1|x− y| 6 |fh(x)− fh(y)| 6 t |x− y|,

(4.10) t−1 |v| 6 |Dfh(x)v| 6 t |v|,

(4.11) t−k 6 Jfh(x) 6 tk,

for every x, y ∈ Eh and v ∈ Rk.

Before the proof of Theorem 4.8 we need the following result.

Exercise: Let f : Rk → Rn be a Lipschitz map. Then there exists a Borel setD ⊂ Rk such that Lk(Rk \D) = 0 and, for each x ∈ D, f is differentiable at x.

(Hint: Use Radamacher’s theorem (see Theorem 3.19 and that Lk is a Borelregular o.m.)

Proof of Theorem 4.8. Assume that

(4.12) Γ = Γ0 ∪ (∪∞i=1f∗i (Ai)) ,

where Hk(Γ0) = 0, Ai ⊂ Rk are Lk-measurable and f ∗i : Ai → Rn are Lipschitzfunctions for i ∈ N. Without loss of generality, by the Borel regularity of Lk, Theorem3.22 and Kirszbraun’s theorem (see Theorem 3.18), we can also assume that, for eachi, Ai is a Borel set and f ∗i : Rk → Rn is a Lipschitz function.

Let us divide the proof in three steps.1st step: We can suppose that, for each i, f ∗i is differentiable at each point x ∈ Ai

and

(4.13) Ai ⊂ Fi :=x ∈ Rk : 0 < Jf ∗i (x) < ∞

.

Indeed, let denote Ai the subset of points x ∈ Ai such that f ∗i is differentiable atx. By the previous exercise we can assume that Ai is a Borel set for each i. Then,since, by Rademacher theorem ),

Lk(Ai \ Ai) = 0 ,

it follows that, by Theorem 3.22,

(4.14) Hk(f ∗i

(Ai \ Ai

))= 0 .

114

By the area formula (see (AF)), it turns out that

(4.15) Hk(f ∗i

(Ai \ Fi

))= 0 for each i .

Therefore, by (4.14) and (4.15), we are allowed to use in (4.12) functions f ∗i : Ai∩Fi →Rn and we get the desired conclusion.

2nd step: By the previous step and applying the Lipschitz linearization to eachfunction f ∗i (see Lemma 3.55), we get that, for each t > 1 there exists a sequence of

disjoint Borel sets (A(i)h )h such that

(4.16) Ai = ∪∞h=1A(i)h ;

(4.17) f ∗i |A(i)h

is injective ;

for each h, there exists a symmetric automorphism Sh ≡ S(i)h : Rk → Rk such that

f ∗i |A(i)hS−1

h : Sh(A(i)h ) ⊂ Rk → f ∗i (A

(i)h ) ⊂ Rn is a bi-Lipschitz map and the following

estimates hold: for every x, y ∈ A(i)h and v ∈ Rn

(4.18) Lip(f ∗i |A(i)

h S−1

h

)6 t, Lip

(Sh (f ∗i |A(i)

h)−1)6 t

(4.19)1

t|Shv| 6 |Df ∗i (x)v| 6 t |Shv|,

(4.20)1

tnJSh 6 Jf(x) 6 tn JSh .

Denote, for each i and h,

G(i)h := Sh

(A

(i)h

)⊂ Rk, g

(i)h := f ∗i |A(i)

h S−1

h : G(i)h → Rn .

Then, by Kirszbraun’s theorem , we can assume that g(i)h : Rk → Rn is still Lipschitz

with Lip(g

(i)h

)6 t and, by Rademacher’s theorem, g

(i)h is differentiable at Lk-a.e.

x ∈ Rk andDg

(i)h (x) = Df ∗i (x) · S−1

h for Lk-a.e. x ∈ G(i)h .

Therefore, by (4.16)- (4.20), and arguing again as in the first step, it follows that

(4.21) (4.8)-(4.11) hold with fh ≡ g(i)h , Eh ≡ G

(i)h .

3rd step: Let us relabel sequences (g(i)h )i,h and (G

(i)h )i,h respectively by sequences

(fh)h and (Eh)h. Therefore, by (4.21), we have that (4.8)-(4.11) hold with fh ≡ fhand Eh ≡ Eh. We have only to show that we can modify respectively the sequenceof functions (fh)h and the one of sets (E∗h)h by a sequence of functions (fh)h and onecompact sets (Eh)h in order that they still satisfies (4.8)-(4.11) and couple (fh, Eh)induces a regular Lipschitz image for each h. For a given h, denote

E(h)m := Eh ∩B(m) if m ∈ N .

where B(m) denotes the closed ball of Rk centered at 0 with radius m.

115

By the approximation of Radon measures by compact sets from below (see Theorem

1.14), for each h, m, there exists an increasing sequence of compact sets (K(h)mj )j such

that

(4.22) K(h)mj ⊂ E(h)

m and Lk(E(h)m \K

(h)mj

)< 1/j ∀ j .

In particular, since

Lk(E(h)m \ ∪∞j=1K

(h)mj

)= 0 , for each h, m ,

by the area formula

(4.23) Hk(fh

(E(h)m \ ∪∞j=1K

(h)mj

))= 0 for each h, m .

Let us denote, for given h, m and j,

f(h)mj := fh and E

(h)mj := K

(h)mj ;

by relabelling sequences (f(h)mj )h,m,j and (E

(h)mj )h,m,j, we get sequences (fh)h and (Eh)h

respectively which still satisfy (4.8)-(4.11). Let us now prove that, for each h, couple(fh, Eh) is a regular Lipschitz image. Observe that each Eh is compact and, by thedensity of a set (see Corollary 2.18) and Lebesgue-Besicovitch differentiation theorem(see Theorem 2.16),we can assume that, for each h and Lk-a.e. x ∈ Eh

limr→0+

Lk(Eh ∩B(x, r))

α∗k rk

= 1, x is a Lebegue point of Dfh .

Thus conditions (ii) and (iii) of Definition 4.8 hold. Moreover, by (4.9) and (4.11),condition (i) of Definition 4.8 holds, too.

4.2. Approximate tangent planes to rectifiable sets. Theorem 4.11 allows usto prove the existence (in a measure-theoretic sense) of tangent spaces to rectifiablesets.

Let us come back to the approximate tangent plane. Define

Φx,r : Rn → Rn as Φx,r(y) :=y − xr

, y ∈ Rn,

so that, if µ is a Radon measure on Rn and E ⊂ Rn is a Borel set, then

(4.24)(Φx,r)#µ(E)

rk=

µ(x+ rE)

rk.

Theorem 4.12 (Existence of approximate tangent spaces). If Γ ⊂ Rn is a locallyHk-rectifiable set, then for Hk-a.e. x ∈ Γ there exists a unique k- dimensional planeπx such that, as r → 0+,

(4.25)(Φx,r)#(Hk Γ)

rk= Hk

(Γ− xr

)∗Hk πx ,

that is

(4.26) limr→0+

1

rk

∫Γ

ϕ

(y − xr

)dHk(y) =

∫πx

ϕ(y) dHk(y) ∀ϕ ∈ C0c(Rn) .

116

In particular

(4.27) ∃Θk(Γ, x) := limr→0+

Hk(Γ ∩B(x, r))

α∗k rk

= 1 for Hk-a.e. x ∈ Γ .

Remark 4.13. Observe that, if (4.25) holds, then πx is unique. Indeed, if (4.25)holds for two k-planes πi (i = 1, 2), by the uniqueness of the local weak* convergenceof measures (see Remark 1.108), then

(4.28) Hk π1 = Hk π2 .

Since spt(Hk πi) = πi (i = 1, 2), by (4.28), π1 = π2.

Definition 4.14 (Approximate tangent plane to a set). Let Γ ⊂ Rn be such thatHk(Γ∩K) <∞ for each compact set K ⊂ Rn. A k-dimensional plane πx is said to bethe approximate tangent plane to Γ at x, if (4.25) holds. Then we denote TxΓ := πx.

Remark 4.15. We will see that the set of points x ∈ Γ such that (4.25) holds truedepends only on the Radon measure µ = Hk Γ. It is a locally Hk-rectifiable set inRn and is unchanged if we modify Γ on and by Hk-null sets (see Theorem 4.21).

Lemma 4.16 (Approximate tangent plane to a regular Lipschitz image). Let f :Rk → Rn be a Lipschitz function and let (f, E) define a regular Lipschitz image inRn. If Γ = f(E), then

(4.29) Tf(z)Γ = df(z)(Rk) Lk-a.e. z ∈ E .

In particular

(4.30) ∃TxΓ = df(f−1(x))(Rk) Hk-a.e. x ∈ Γ .

Before the proof of Lemma 4.16, let us point out the following

Exercise: Let f : Rk → Rn be differentiable at z ∈ Rk and assume thatJf(z) > 0. Then there exist λ, s0 > 0 such that

(4.31) |f(z′)− f(z)| > λ |z′ − z| ∀ z′ ∈ B(z, s0) .

(Hint: First observe that, since Jf(z) > 0, rank(df(z)) = k. This implies that

λ0 := min|df(z)(v)| : v ∈ Rk, |v| = 1

> 0.

On the other hand, since f is differentiable at z, one can prove that, if λ = λ0/2there exits s0 > 0 such that (4.31) holds. )

Proof. If ϕ ∈ C0c(Rn), then by the integration with respect to a push- forward measure

(see (1.69)) and the change of variables (see Theorem 3.56) we have

1

rk

∫Rnϕd((Φx,r)#(Hk Γ)

)=

1

rk

∫Γ

ϕ Φx,r dHk =1

rk

∫Γ

ϕ

(y − xr

)dHk(y)

=1

rk

∫E

ϕ

(f(w)− f(z)

r

)Jf(w) dLk(w)

=

∫Rkur(w) dLk(w) ,

117

where ur : Rk → R is defined as

ur(w) := χE(z + rw)ϕ

(f(z + rw)− f(z)

r

)Jf(z + rw) .

Let (rh)h be an arbitary sequence of positive numbers satisfying limh→∞ rh = 0. Letz be a Lebesgue point of χE and Jf , and f is differentiable at z, it is easy to seethat, up to a subsequence, for each R > 0

limh→∞

urh(w) = u0(w) := ϕ (df(z)(w)) Jf(z) Lk-a.e. w ∈ B(R) .

and then

(4.32) limh→∞

urh(w) = u0(w) Lk-a.e. w ∈ Rk .

It is also easy to check that

(4.33) |ur(w)| 6 supRn|ϕ|Lip(f)k Lk-a.e. w ∈ Rk, ∀ r > 0 .

We are going now to prove that there exist r0 and R0 > 0 such that

(4.34) spt(ur) ⊂ B(R0) ∀ r ∈ (0, r0) .

Indeed, from (4.32)-(4.34), the dominated convergence theorem and the area formula,it will follow that

limh→∞

1

rkh

∫Γ

ϕ Φx,rh dHk = limh→∞

∫Rkurh(w) dLk(w) =

∫Rku0(w) dLk(w)

=

∫Rkϕ (df(z)(w)) Jf(z) dLk(w) =

∫df(z)(Rk)

ϕdHk

for each arbitrary sequence (rh)h of positive real numbers with limh→∞ rh = 0. Thus(4.29) is proved. Finally, let us show (4.34). By the previous exercise, there existλ, s0 > 0 such that

(4.35) |f(z′)− f(z)| > λ |z′ − z| ∀ z′ ∈ B(z, s0) .

Moreover, if R > 0 is such that spt(ϕ) ⊂ B(R), then

(4.36) |f(z + rw)− f(z)| 6 r R ∀w ∈ spt(ur) .

By the compactness of E and injectivity of f on E one has

inf |f(z′)− f(z)| : z′ ∈ E \ U(z, s0) = ε0 > 0,

so that, by (4.35), one gets

(4.37) |f(z′)− f(z)| > min

λ,

ε0d(E)

|z − z′| = c0 |z − z′| ∀ z′ ∈ E .

In this way, if w ∈ spt(ur) then z + rw ∈ E and, by (4.36), |f(z + rw)− f(z)| 6 Rr,so that, by (4.37) c0 r |w| 6 Rr. This proves spt(ur) ⊂ B(R0) with R0 = R/c0, whichshows (4.34). Eventually, Let N ⊂ E denote the set of points z ∈ E such that z isnot a Lebesgue point of χE. Then Lk(N)= 0. Thus, by the area formula

Hk(f(N)) = 0 .

Thus, by (4.29), for each x ∈ Γ \ f(N), (4.32) holds.

118

Proof of Theorem 4.12. 1st step. We decompose Γ = Γ0 ∪∞h=1 f(Eh) as in Theorem4.11. If we let Γh = fh(Eh), then by Lemma 4.16 we find that

(4.38) limr→0+

1

rk

∫Γh

ϕ Φx,r dHk =

∫πx

ϕdHk ∀ϕ ∈ C0c(Rn), Hk − -a.e. x ∈ Γh ,

where we have set πx := dfh(f−1h (x))(Rk). Since Hk Γ is a Radon measure on Rn,

by Corollary 3.38,

(4.39) Θk(Γ \ Γh, x) = limr→0+

Hk((Γ \ Γh) ∩B(x, r))

α∗k rk

= 0 Hk-a.e. x ∈ Γh .

For a given ϕ ∈ C0c(Rn), suppose that spt(ϕ) ⊂ B(R0). Then

1

rk

∣∣∣∣∫Γ\Γh

ϕ Φx,r dHk

∣∣∣∣ =1

rk

∣∣∣∣∫Γ\Γh

ϕ

(y − xr

)dHk(y)

∣∣∣∣=

1

rk

∣∣∣∣∫(Γ\Γh)∩B(x,R0r)

ϕ

(y − xr

)dHk(y)

∣∣∣∣6 sup

Rn|ϕ| H

k ((Γ \ Γh) ∩B(x,R0r))

rk.

Thus, by (4.39), it follows that

limr→0+

1

rk

∫Γ\Γh

ϕ Φx,r dHk

= limr→0+

1

rk

∫Γ\Γh

ϕ

(y − xr

)dHk(y) = 0 Hk-a.e. x ∈ Γh .

(4.40)

From (4.38) and (4.40), (4.26) follows.2nd step. Let x ∈ Γ satisfy (4.25). Since, by (4.5), Hk(π ∩ ∂B1) = 0, by Theorem1.113, we can infer that

α∗k = Hk(π ∩B1) = limr→0+

Hk Γ(Φ−1x,r(B1)

)rk

= limr→0+

Hk(Γ ∩B(x, r))

rk,

and thus (4.27) follows.

Let us also point out the following interesting locality result for the approximatetangent plane.

Proposition 4.17 (Locality of the approximate tangent plane). If Γi (i = 1, 2) arelocally Hk-rectifiable sets of Rn, then for Hk-a.e. x ∈ Γ1 ∩ Γ2

TxΓ1 = TxΓ2 .

Proof. See [Mag, Proposition 10.5].

Example 4.18 (Approximate tangent space to a Lipschitz (n − 1)-graph). If φ :Rn−1 → R is a Lipschitz function, and we define f : Rn−1 → Rn as f(z) = (z, φ(z)),z ∈ Rn−1, then, as pointed out in Example 4.6, Γ := f(Rn−1) = graph(φ) is locallyHn−1-rectifiable and. Let us now show that for Ln−1-a.e. z ∈ Rn−1, if ν(z) :=(−∇φ(z), 1),

(4.41) Tf(z)Γ = ν(z)⊥ := v ∈ Rn : (v, ν(z))Rn = 0 ,

119

or, equivalently, for Hn−1-a.e. x ∈ Γ,

(4.42) TxΓ = ν(f−1(x))⊥ .

In particular this implies that the classical tangent space to a Lipschitz (n − 1)-dimensional graph agrees Hn−1-a.e. with the approximate tangent space.

Let us first observe that f is Lipschitz, injective and it actually satisfies

(4.43) |z − z′| 6 |f(z)− f(z′)| 6√

1 + Lip(φ)2 |z − z′| ∀ z, z′ ∈ Rn−1 .

Moreover, by (3.73),

(4.44) Jf(z) =√

1 + |∇φ(z)|2 Ln−1-a.e. z ∈ Rn−1

and, by the area formula,

(4.45) Hn−1 Γ(E) =

∫f−1(Γ∩E)

√1 + |∇φ(z)|2dLn−1

for each Borel set E ⊂ Rn.Let A1 ⊂ Rn−1 denote the set of points x satysfying:

• φ is differentiable at x;• x is a Lesbegue point of ∂jφ for each j = 1, . . . , n − 1 (with respect to the

(n− 1)-dimensional Lebesgue measure Ln−1);

and let A2 ⊂ A1 denote the set of points of density 1 in A1 (with respect to the(n− 1)-dimensional Lebesgue measure Ln−1). By Rademacher’s and Lebesgue pointtheorems (see Theorems 3.19 and 2.1) and the density of a set (see Corollary 2.18)Ai (i = 1, 2) are measurable and they have full measure, that is Ln−1(Rn−1 \Ai) = 0(i = 1, 2). Let us define

AR := A2 ∩B(R) and ΓR := f(AR) if R > 0 .

Since AR is measurable and Ln−1(AR) <∞, by the approximation of Radon mea-sures by means of compact sets (see Theorem 1.14), we can find and increasing se-quence of compact sets (Eh)h such that

(4.46) Eh ⊂ AR ∀h and Ln−1(AR \ ∪∞h=1Eh) = 0 .

LetΓR,0 := f(AR \ ∪∞h=1Eh), ΓR,h := f(Eh) .

Then, by construction and (4.45),

ΓR = ΓR,0 ∪ (∪∞h=1ΓR,h) and Hn−1(ΓR,0) = 0 ;

by definiton of AR, (4.43), (4.44), and (4.45)

ΓR,h = f(Eh) is a regular Lipschitz image for each h .

Applying Lemma 4.16, for a given h, it follows that

Tf(z)ΓR,h = df(z)(Rn−1) Ln−1-a.e. z ∈ Eh.By standard linear algebra, we can infer that

df(z)(Rn−1) = span v1(z), . . . , vn−1(z) = ν(z)⊥ Ln−1-a.e. z ∈ Eh ,with

v1(z) := e1 + ∂1φ(z) en, . . . , vn−1(z) := en−1 + ∂n−1φ(z) en ,

120

where e1, . . . , en denotes the standard basis of Rn. It is easy to see that vectorsvi(z) ∈ ν(z)⊥ for each z ∈ Eh, i = 1, . . . , n − 1 and they are a basis of the subspaceν(z)⊥. Thus, we have proved that

Tf(z)ΓR,h = ν(z)⊥ Ln−1-a.e. z ∈ Eh.

or that is equivalent

(4.47) TxΓR,h = ν(f−1(x))⊥ Hn−1-a.e. x ∈ ΓR,h .

By the locality of the approximate tangent plane (see Proposition 4.17) and (4.47),it follows that, for every h,

TxΓ = TxΓR,h = ν(f−1(x))⊥ Hn−1-a.e. x ∈ ΓR,h = ΓR,h ∩ Γ ,

and then

(4.48) TxΓ = ν(f−1(x))⊥ Hn−1-a.e. x ∈ ΓR = ΓR ∩ Γ .

By (4.48), since R > 0 is arbitrary, (4.42) follows. Finally, by (4.45), (4.42) and(4.41) are equivalent.

Let us point out the striking fact that (4.27) implies the rectifiability. Indeed

Theorem 4.19 (Besicovitch-Marstrand-Mattila). Let E a Borel set with Hk(E) <∞. Then the following are equivalent:

(i) E is Hk-rectifiable;(ii) there exists Θk(E, x) = 1 for Hk-a.e. x ∈ E.

Proof. Implication (i)⇒(ii) follows from Theorem 4.12. Implication (ii)⇒(ii) is muchharder and can be found in [Ma, Theorem 17.6].

Remark 4.20. Preiss improved these results in [Pre2] proving that the existence ofΘk(E, x) ∈ (0,∞) implies the Hk-rectifiabilty of E.

4.3. Blow-ups of Radon measures on Rn and rectifiability. We now prove aconverse statement to Theorem 4.12, which plays an important role in GMT and, inparticular, when studying the structure of sets of finite perimeter (see Chap. V).

Theorem 4.21 (Rectifiability by convergence of the blow-ups). If µ is a Radonmeasure on Rn, Γ is a Borel set in Rn, µ is concentrated on Γ (that is µ = µ Γ),and, for every x ∈ Γ, there exists a k-dimensional plane πx in Rn such that

(Φx,r)#µ

rk∗Hk πx as r → 0+,

then µ = Hk Γ and Γ is locally Hk-rectifiable.

The proof of Theorem 4.21 relies on a simple criterion for Hk rectifiability, veryuseful in GMT.

121

Definition 4.22 (Cones). If π is a k-plane, define the cone K(π, t) and openingt > 0, as

K(π, t) : =y ∈ Rn : |P⊥π (y)| 6 t |Pπ(y)|

=y ∈ Rn : |y| 6

√1 + t2 |Pπ(y)|

=

y ∈ Rn : d(y, π) 6

√1

1 + t2|y|

(4.49)

Notice that K(π, t) is invariant by dilations, that is sK(π, t) = K(π, t) for eachs > 0, K(π, t) reduces to π if t = 0, and K(π, t) \ 0 ↑ Rn \ π⊥ as t ↑ ∞.

Subsets of Lipschitz k-graphs can be easily characterised by cones. In fact

Exercise: Let S ⊂ Rn. Then the following are equivalent:

(i) there exist a k-dimensional plane π and an opening t > 0 such that S ⊂x+ K(π, t) for each x ∈ S;

(ii) there exist a k-dimensional plane π , a unique function φ : E ⊂ π → π⊥ ⊂ Rn

and a constant t > 0 such that φ is t-Lipschitz and graph(φ) := z + φ(z) :z ∈ E = S.

(Hint: Implication (ii)⇒ (i) is trivial. Implication (i)⇒ (ii) follows by noticing that:a) if x1, x2 ∈ S and Pπ(x1) = Pπ(x2) then x1 = x2;b) for each z ∈ E := Pπ(S) ⊂ π there exists a unique y := φ(z) ∈ π⊥ such that

z + y ∈ S and φ : E ⊂ π → π⊥ ⊂ Rn is a t-Lipschitz function. )

Theorem 4.23 (Rectifiability criterion). If Γ ⊂ Rn is a compact set, π is a k-dimensional plane in Rn, and there exist δ and t positive with

(4.50) Γ ∩B(x, δ) ⊂ x+K(π, t) ∀x ∈ Γ ,

then Γ is Hk-rectifiable, since there exist finitely many Lipschitz maps fh : Rk → Rn

(h = 1, . . . , N) and compact sets Fh ⊂ Rk with

Γ = ∪Nh=1fh(Fh) .

Proof. Let z ∈ Γ, then notice that

(4.51) Γ ∩B(z, δ/2) ⊂ x+K(π, t) ∀x ∈ Γ ∩B(z, δ/2) .

Since Γ is compact, there exist x1, . . . , xN ∈ Γ such that Γ ⊂ ∪Nh=1B(xh, δ/2) and(4.51) holds with z = xh h = 1, . . . , N . By the previous exercise, for each h =1, . . . , N there exists a t-Lipschitz function

φh : Fh := Pπ (Γ ∩B(xh, δ/2)) ⊂ π → π⊥ ⊂ Rn

such that

(4.52) graph(φh) = Γ ∩B(xh, δ/2) ,

and

Γ = ∪Nh=1 (Γ ∩B(xh, δ/2)) = ∪Nh=1graph(φh) ,

with Fh ⊂ π (h = 1, . . . , N) compact sets. Let us define, for h = 1, . . . , N ,

gh : Fh ⊂ π → Rn, gh(z) := z + φh(z) .

122

Then gh is Lipschitz and, by Corollary 3.17, we can assume that gh is defined on thewhole π. Let us now define

fh := gh Iπ : Rk → Rn

where Iπ is the map defined in (4.1), we get the desired conclusion by choosingFh := I−1

π (Fh) (h = 1, . . . , N).

Remark 4.24. As a consequence of (4.52), we also proved that a compact set Γ ⊂Rn satisfying (4.50) is the union of finitely many Lipschitz k-graphs, according toExample 4.6 and then a Hk- rectifiable set.

We will also need a technical lemma concerning inclusions between cones and balls.

Lemma 4.25. There is s0 ∈ (0, 1) such that for each two k-dimensional planes πand σ satisfying |π − σ|G < s0, it holds that

(4.53) B(w, s0|w|) ∩K(π, 1) = ∅ ∀w ∈ Rn \K(σ, 2) .

Proof. Let us begin to observe that, since sets U(w, s0|w|), K(π, 1) and Rn \K(σ, 2)are invariant by dilations, without loss of generality, we can prove (4.53) for eachw ∈ Sn−1 ∩ (Rn \K(σ, 2)). By contradiction, assume there are a sequence (sh)h ↓ 0,two sequences of k-dimensional planes (πh)h and (σh)h and two sequences (wh)h and(zh)h in Rn satisfying, for each h,

(4.54) |πh − σh|G < sh ,

(4.55) zh ∈ B(wh, sh) ∩K(πh, 1) ⇐⇒ |wh − zh| 6 sh and |zh| 6√

2 |Pπh(zh)| ,

(4.56) wh ∈ Sn−1 ∩ (Rn \K(σh, 2)) ⇐⇒ |wh| = 1 and |wh| >√

5 |Pσh(wh)| .By compactness (recall (4.3)), we can assume that there exist two k-dimensionalplanes π∗ and σ∗, a pont w∗ ∈ Sn−1 such that

(4.57) limh→∞|πh − π∗|G = lim

h→∞|σh − σ∗|G = 0 ,

(4.58) limh→∞|w − w∗| = 0 .

Thus, by (4.54)-(4.58), we can infer that

(4.59) π∗ = σ∗ and ∃ limh→∞

zh = w∗ .

By the definition of distance | · |G and (4.60), the sequences of operators (Pπh)h and(Pσh)h uniformly converge on Sn−1 to Pπ∗ . In particular we also get that

(4.60) limh→∞|Pπh(zh)− Pπ∗(w∗)| = lim

h→∞|Pσh(wh)− Pπ∗(w∗)| = 0 .

By (4.60), we can pass to the limit as h → ∞ in the two inequalities in (4.55) and(4.57) and we get √

5 |Pπ∗(w∗)| 6 1 6√

2 |Pπ∗(w∗)| ,and then a contradiction.

An other useful tool in GMT is the following version of Severini-Egoroff’s theorem1.18 which applies to Radon measures on Rn and a family of convergent functionsrather than a sequence of convergent functions.

123

Lemma 4.26 (Severini-Egoroff’s theorem for a convergent family of functions). Letϕ be a finite Radon outer measures on Rn, let Γ ⊂ Rn be ϕ-measurable and letfr : Γ→ R (r > 0) be a family of functions such that

∃ limr→0+

fr(x) = f0(x) ϕ-a.e. x ∈ Γ .

Then, for each ε > 0, there exists a compact set Γ′ ⊂ Γ such that

ϕ(Γ \ Γ′) < ε ,

and

fr → f0 uniformly on Γ′ as r → 0+, i.e. limr→0+

supx∈Γ′|fr(x)− f(x)| = 0 .

Proof. Let us define the sequence of functions gh : Γ→ Rgh(x) := sup |fr(x)− f(x)| : 0 < r < 1/h h = 1, 2, . . . .

Observe that gh (h = 1, 2, . . . ) is well- defined, ϕ- measurable and

limh→∞

gh(x) = 0 µ-a.e. x ∈ Γ .

Indeed, it is easy to see that

gh(x) := sup |fr(x)− f(x)| : r ∈ Q, 0 < r < 1/h .thus, since the family fr : r > 0 is ϕ-measurable, also gh is ϕ-measurable. Givenε > 0, applying Severini-Egoroff’s theorem 1.18 to sequence (gh)h we get the existsnceof a ϕ-measurable set E ⊂ Γ such that

(4.61) ϕ(Γ \ E) < ε/2 and gh → 0 uniformly on E as h→∞ .

On the other hand, by the approximation theorem for Radon measures (see Theorem1.14), there is a compact set Γ′ ⊂ E such that

(4.62) ϕ(E \ Γ′) < ε/2 .

Therefore, by (4.61) and (4.62), we get the desired conclusion.

Proof of Theorem 4.21. Let s0 be the positive constant in Lemma 4.25. Then, by(4.3), there is finite family of k-dimensional planes σ1, . . . , σN such that

(4.63) min16h6N

|σh − π|G 6s0

2for each k-dimensional plane π .

Let us now divide the proof in three steps.1st step. We show that if Γ′ is a compact subset of Γ and assume that the limitrelations

limr→0+

µ(B(x, r))

α∗k rk

= 1 ,

limr→0+

µ (B(x, r) \ (x+K(πx, 1)))

α∗k rk

= 0 ,

hold uniformly with respect to x ∈ Γ′, then Γ′ is Hk-rectifiable. Thus, assume that:for each ε > 0 there exists δ > 0 such that for each x ∈ Γ′, r ∈ (0, δ)

(4.64) 1− ε < µ(B(x, r))

α∗k rk

< 1 + ε ,

124

(4.65) µ (B(x, r) \ (x+K(πx, 1))) < εα∗k rk .

Let

Γ′h :=x ∈ Γ′ : |σh − πx|G 6

s0

2

1 6 h 6 N .

Then let us show that

(4.66) Γ′h ∩B(x, δ) ⊂ x+K(σh, 2) ∀x ∈ Γ′h, h = 1, . . . , N .

By contradiction, if x ∈ Γ′h and y ∈ B(x, δ) ∩ Γ′h \ x but y /∈ (x + K(σh, 2)), thatis y − x ∈ Rn \K(σh, 2), then, by (4.53),

(4.67) B(y, s0|y − x|) ⊂ Rn \ (x+K(πx, 1)) .

Since s0 ∈ (0, 1)

B(y, s0|y − x|) ⊂ B(x, 2 |y − x|)by (4.67), it follows that

B(y, s0|y − x|) ⊂ B(x, 2 |y − x|) \ (x+K(πx, 1)) .

Applying (4.65) (at x with r = 2|y − x|) and (4.64) (at y with r = s0|y − x|)

ε α∗k 2k |y − x|k > (1− ε)α∗k sk0 |y − x|k

a contradiction, as soon as ε is small enough with respect to k and s0. This proves(4.66). By the rectifiability criterion ( Theorem 4.23) , Γ′h is thus Hk-rectifiable forh = 1, . . . , N . By (4.63), Γ′ ⊂ ∪Nh=1Γ′h. Thus Γ′ is Hk -rectifiable.2nd step. Let us prove that Γ is countable Hk-rectifiable. We have

(4.68) limr→0+

µ(B(x, r))

α∗k rk

= 1 ,

(4.69) limr→0+

µ (B(x, r) \ (x+K(πx, 1)))

α∗k rk

= 0

for each x ∈ Γ. Indeed, since, by (4.5), Hk π(∂B1) = Hk(π ∩ ∂B1) = 0, therefore,by Theorem 1.113,

α∗k = Hk(πx ∩B1) = limr→0+

(Φx,r)#µ(B(x, r))

rk= lim

r→0+

µ((B(x, r))

rk,

that is (4.68). Let us now check (4.69) in a similar way. Notice that, if E :=B1 \K(πx, 1), then

x+ r E = B(x, r) \ (x+K(πx, 1)) ,

and

πx ∩ E = ∅, πx ∩ ∂E = 0 .Thus, since Hk π(∂E) = 0, by (4.24),

0 = Hk(π ∩ E) = limr→0+

(Φx,r)#µ(E)

rk= lim

r→0+

µ (B(x, r) \ (x+K(πx, 1)))

α∗k rk

.

Let us now prove tha, for each Borel set E ⊂ Rn

(4.70) Hk Γ(E) 6 µ(E) = µ Γ(E) 6 2kH Γ(E) .

125

Indeed, by the estimates of the upper density of a Radon measure (see Theorem 3.37)since, by (4.68),

1 6 Θk(µ, x) = Θ∗k(µ, x) 6 1 ∀x ∈ Γ ,

(4.70) follows.Given R > 0, since µ(B(R)) < ∞, by applying Lemma 4.26 with ϕ = µ B(R),

we can infer the existence of a compact set Γ1 ⊂ Γ such that the limit relations (4.68)and (4.69) hold uniformly on Γ1 with

µ((Γ ∩B(R)) \ Γ1) <µ(Γ ∩B(R))

2.

By an induction procedure, we can construct a disjoint sequence of compact sets (Γh)hsuch that

(4.71) µ ((Γ ∩B(R) \ (∪mh=1Γh)) <µ(Γ ∩B(R))

2m∀m = 1, 2, . . . .

(4.72) the limit relations (4.68) and (4.69) hold uniformly on Γh

By step 1 and (4.72), it follows that Γh (h = 1, 2, . . . ,) is Hk- rectifiable. Since, by(4.71), µ((Γ∩B(R))\(∪∞h=1Γh)) = 0, by (4.70) we can also infer that Hk((Γ∩B(R))\(∪∞h=1Γh)) = 0. Thus, by definition,

Γ ∩B(R) is countably Hk- rectifiable for each R > 0 ,

and since Γ = ∪∞h=1(Γ ∩B(h)), we also get that

Γ is countably Hk- rectifiable .

Let us now prove that Γ is actually locally Hk- rectifiable. By (4.70), it follows that,since µ is a Radon o.m., Hk(Γ ∩K) < ∞ for each compact set K ⊂ Rn. Thereforewe get the desired conclusion.3rd step. Let us prove that µ = Hk Γ on P(Rn). Since, by (4.70), Hk Γ is aRadon o.m., absolutely continuous with respect to µ, by the differentiation theoremfor positive Radon measures (see Theorem 2.15), Hk Γ = w dµ with

(4.73) w(x) = DHk Γµ(x) = limr→0+

µ(B(x, r))

Hk(Γ ∩B(x, r))Hk-a.e.x ∈ Γ .

By (4.26), (4.68) and (4.73), it follows that w(x) = 1 for Hk-a.e. x ∈ Γ. ThusHk Γ = µ on the class of Borel sets. By Remark 1.15, it follows that they agree onP(Rn).

Purely unrectifiable sets

Definition 4.27 (Purely unrectifiable sets). Let E ⊂ Rn be a Borel set. We saythat E is purely Hk-unrectifiable if Hk(E ∩ f(Rk)) = 0 for any Lipschitz functionf : Rk → Rn.

126

Equivalently, we might say that E is purely Hk-unrectifiable if Hk(E ∩ F ) = 0 forany countably Hk-rectifiable set F . There exist examples of purely Hk-unrectifiablewith Hausdorff dimension strictly greater than k (for k = 1 one example is the vonKoch snowflake in the plane, whose Hausdorff dimension is log 4/ log 3, see [Ma, 4.13]and [Fa, Introduction]). In the next example we show how a purely H1- unrectifiableset in R2 with Hausdorff dimension 1 can be constructed.

Example 4.28 ( An unrectifiable set). Let C := C(1/4) be the Cantor set definedin (3.51) with λ = 1/4 and let E := C ×C ⊂ R2 (called the Cantor dust in the unitsquare). It is easy to see that E still presents a self-similar structure. Indeed, let

Ck := Ck(1/4) = ∪2k

j=1Ik,j k = 1, 2, . . . ,

be the sets defined in (3.51) for C := C(1/4)’s definition. Since, by definition,

C := ∩∞k=1Ck ,

then

E := C × C = ∩∞h=1(Ck × Ck) .with

Ck × Ck =(∪2k

j=1Ik,j

)×(∪2k

l=1Ik,l

)= ∪2k

i,l=1 (Ik,j × Ik,l) = ∪2k

i,l=1Qk,jl

(4.74)

where Qk,jl j, l = 1, . . . , 2k, is a family of 4k closed disjoint squares with side length4−k for k = 1, 2, . . . .

Then we claim that:

(i) 0 < 3/√

5 6 H1(E) 6√

2.

(ii) Θ1∗(E, (x, y)) := lim inf

r→0+

H1(E ∩B((x, y), r)

2 r6

1

2< 1 for each (x, y) ∈ E.

(iii) Let Ph denote the union of the boundaries of 4h squares Qh,jl for j, l =1, . . . , 2h. Then it holds that (H1 Ph)h weakly* converges to cH1 E onB(R2) as h→∞ with c = 4/H1(E).

From (i), it follows that Hdim(E) = 1 and, by Theorem 4.12 and (ii), that Eis purely H1-unrectifiable. Claim (iii) shows that weak* convergence is an approx-imation too weak to preserve the concept of rectifiability. Indeed sets Ph are finiteunions of Lipschitz curves and then H1-rectifiable, instead of E, which is purely H1-unrectifiable. Finally notice that E is totally disconnected, that is its connectedcomponents are points. Indeed it can be proved that, if Γ ⊂ Rn is closed and con-nected with H1(Γ) < ∞, then Γ is H1-rectifiable (see Theorem 4.31).

Let us now prove (i),(ii) and (iii).

Proof of (i): From (4.99), it follows that, if δk :=√

2 4−k,

H1δk

(E) 6√

2 ∀ k

and so inequality H1(E) 6√

2 follows. We have now to prove that

(4.75)3√56 H1(E) .

127

Letπ :=

(x, y) ∈ R2 : y = 2x

and let Pπ : R2 → R2 denote the orthogonal projection of R2 on straight line π. If Tis the segment of π defined by

T :=

(x, y) ∈ R2 : y = 2x, 0 6 x 6

3

5

,

then T turns out to be the orthogonal projection on π both of square Q0,1 and strip

S0,1 :=

(x, y) ∈ R2 : −x

26 y 6 −x

2+

3

2

that is

(4.76) Pπ (S0,1) = Pπ (Q0,1) = T ,

which is quite evident since the orthogonal subspace to π turns out to be

π⊥ =

(x, y) : y = −x2

.

We also claim that

(4.77) Pπ(E) = T .

Since

H1 (T ) =3√5

and Pπ : R2 → R2 is a 1- Lipschitz map, by (4.77) and Propositon 3.22,

3√5

= H1 (T ) = H1 (Pπ(E)) 6 H1(E) ,

and (4.75) follows. Thus we have only to show (4.77).Thus it suffices to prove that

(4.78) Pπ(Ck × Ck) = T for each k = 1, 2, . . . ,

in order to show (4.77). Let us define, for k = 1, 2, . . . , the family of 4k substrips ofS0,1

Sk,h :=

(x, y) ∈ R2 : −x

2+h− 1

4k3

26 y 6 −x

2+h

4k3

2

h = 1, 2, . . . , 4k ,

It is easy to prove that,

(4.79) S0,1 = ∪4k

h=1Sk,h for each k = 1, 2, . . . ,

and, for a given k = 1, 2, . . . , for each integer 1 6 h 6 4k there are unique integers1 6 jh 6 2k and 1 6 lh 6 2k such that

(4.80) Qk,h := Qk,jhlh ⊂ Sk,h and Pπ(Qk,h) = Pπ(Sk,h) .

Thus we can relabelling the family of 4k squares Qk,jl j, l = 1, . . . , 2k by means ofthe family Qk,h, h = 1, 2, . . . , 4k, such that

(4.81) Ck × Ck = ∪4k

h=1Qk,h ,

128

(4.82) d(Qk,h, Qk,m) >1

2 4k−1if h 6= m.

By (4.87), (4.79), (4.80) and (4.81), we can infer that

Pπ(Ck × Ck) = Pπ

(∪4k

h=1Qk,h

)= ∪4k

h=1Pπ(Qk,h)

= ∪4k

h=1Pπ(Sk,h) = Pπ

(∪4k

h=1Sk,h

)= Pπ(S0,1) = T

and (4.78) follows.

Proof of (ii): Let us begin to point out the self-similar structure of E . Let λ = 1/4

and define the following four similarities Si : R2 → R2 (i = 1, 2, 3, 4)

S1(x, y) := (λx, λ y), S2(x, y) := S1(x, y) + (1− λ, 0)

S3(x, y) := S1(x, y) + (0, 1− λ), S4(x, y) := S1(x, y) + (1− λ, 1− λ) .

Since

(4.83) Si(Q0,1) = Q1,i for i = 1, 2, 3, 4 ,

it is easy to check that

(4.84) ∪4i=1Si(Q0,1) = ∪4

i=1Q1,i = C1 × C1

and

(4.85) ∪4i=1Si(Ck × Ck) = Ck+1 × Ck+1 for each k > 1 .

In particular it follows that

(4.86) ∪4i=1Si(E) = E ,

and

(4.87) Si(E) = E ∩Q1,i for i = 1, 2, 3, 4 .

Moreover, as usual in a self-similar uniform structure, we can infer that

(4.88) H1(E ∩Qk,i) =1

4kH1(E) for k > 1, i = 1, 2, . . . , 4k .

Indeed, by induction on k, (4.88) holds for k = 1 by (4.83) and (4.86). Assume that(4.88) for k, then let us prove it hold for k + 1. Fix an integer 1 6 s 6 4k+1. Thenit is clear that there exist integers 1 6 i∗ 6 4 and 1 6 s∗ 6 4k such that

Qk+1,h = Si∗ (Qk,s∗) .

Therefore, by (4.83), (3.24), (3.25) and the inductive hypothesis,

H1(E ∩Qk+1,s) = H1 (E ∩ Si∗ (Qk,s∗)) = H1 (E ∩Q1,i∗ ∩ Si∗ (Qk,s∗))

= H1 (Si∗(E) ∩ Si∗ (Qk,s∗)) =1

4H1 (E ∩Qk,s∗) =

1

4k+1H1 (E) .

Fix (x, y) ∈ E. By definition, (x, y) ∈ Ck×Ck for each k = 1, 2, . . . , then there existsa unique integer s = s(x, y) ∈ 1, 2, . . . , 4k such that

(4.89) (x, y) ∈ Qk,s if and only if h = s ,

129

and, if rk :=√

2 4−k,

(4.90) E ∩Qk,s = E ∩B((x, y), rk) ∀ k = 1, 2, . . . .

Indeed because of (4.81) and (4.82), the family of cubes Qk,s s = 1, 2, . . . , 4k coveringCk × Ck are disjoint, which yields (4.89). Since d(Qk,s) 6 rk, it is immediate that

E ∩Qk,s ⊆ E ∩B((x, y), rk) .

The reverse inclusion follows by noticing that

B((x, y), rk) ∩Qk,s = ∅ for each s 6= s ,

otherwise,

d(Qk,s, Qk,s) 6 rk =

√2

4k<

1

2 4k−1with s 6= s ,

which contradicts (4.82). This proves (4.90). By (4.90), (4.88) and the previous claim(i), it follows that

H1(E ∩B((x, y), rk)

2 rk=H1(E ∩Qk,s)

2 rk6H1(E)

2√

26

1

2∀ k ,

which implies

lim infr→0+

H1(E ∩B((x, y), r)

2 r6 lim inf

k→∞

H1(E ∩B((x, y), rk)

2 rk6

1

2,

and the proof is accomplished.

Proof of (iii): Let µh := H1 Ph : B(R2) → [0,∞] and λ := cH1 E : B(R2) →[0,∞]. Then, since

µh(R2) = 4 ∀h ,by the compactness of weak* convergence (see Theorem 1.121)), it is not restrictiveto assume that there exists a Radon measure µ on R2 such that

(4.91) (µh)h weakly* converges to µ .

In particular we also get that µh∗µ as h → ∞ and, by the characterization of the

local weak* convergence (see Theorem 1.113),

(4.92) µ(K) > lim suph→∞

µh(K) for each compact set K ,

(4.93) µ(A) 6 lim infh→∞

µh(A) for each open set A .

We are going to show that

(4.94) µ(F ) = λ(F ) for each Borel set F ⊂ R2 .

Let us begin to show that µ is concentrated on E, that is

(4.95) µ(R2 \ E) = 0 and spt(µ) = E .

Observe that, by definition, since Ph ⊂ Ck × Ck if h > k > 1, then

(4.96) Ph ∩ F = Ph ∩ (∩mk=1Ck × Ck) ∩ F ∀h > m > 1, F ⊂ R2 ,

and, for each h > k > 1, s = 1, . . . , 4k

(4.97) 4 = H1 (Ph) = H1 (Ph ∩ (Ck × Ck)) = 4kH1 (Ph ∩Qk,s)

130

By (4.92), (4.93) and (4.96), it follows that, for each compact set K ⊂ R2 and integerm,

µ(K) 6 lim inf

h→∞H1 Ph(

K) 6 , lim sup

h→∞H1 Ph(K)

= lim suph→∞

H1 Ph (K ∩ (∩mk=1Ck × Ck))

6 µ (K ∩ (∩mk=1Ck × Ck)) .Passing to the limit, when m→∞, in the previous inequality, we get that

µ(K) 6 µ (K ∩ E) for each compact set K ⊂ R2 ,

and, since E is compact, this implies that

µ(R2) 6 µ(E) < ∞and then (4.95) follows. Let us now prove that

(4.98) µ (B((x, y), rk)) = µ (E ∩B((x, y), rk)) =1

4k−1for each k > 1, (x, y) ∈ E .

The first identity immediately follows by (4.95). By (4.89), (4.90) and (4.95), we caninfer that, for each (x, y) ∈ E, for each k there exists a unique 1 6 s 6 4k such that

(4.99) µ (E ∩B((x, y), rk)) = µ (E ∩Qk,s) = µ (Qk,s) .

By (4.92) and (4.97), it follows that

µ (Qk,s) > lim suph→∞

H1 (Ph ∩Qk,s) = 41−k .

The reverse inequality can be obtained noticing that, by (4.82), we can fatten theclosed cube Qk,s by an open cube Qk,s in such a way, for each s = 1, . . . , 4k,

Qk,s ⊂ Qk,s and Qk,s ∩Qk,s 6= ∅ if and only if s = s .

Thus, by (4.95), (4.93) and (4.97),

µ (Qk,s) = µ (E ∩Qk,s) = µ(E ∩ Qk,s

)= µ

(Qk,s

)6 lim inf

h→∞µh

(Qk,s

)= lim inf

h→∞µh (Qk,s) = 41−k .

By the two previous inequalities and (4.99), the second identity in (4.96) also follows.By (4.98), (4.88) and (4.90), we can infer that

µ (B((x, y), rk))

= λ (B((x, y), rk)) = 41−k ∀ (x, y) ∈ spt(µ) = E, k = 1, 2, . . . .(4.100)

Let us now prove that

(4.101) µ << λ

or, that is equivalent by the differentiation for positive measures (see Theorem 2.15),

(4.102) Dλµ(x, y) := lim infr→0+

µ (B((x, y), r))

λ (B((x, y), r))< ∞ µ-a.e. (x, y) ∈ R2 .

131

By (4.100), we immediately get that

lim infr→0+

µ (B((x, y), r))

λ (B((x, y), r))6 lim inf

k→∞

µ (B((x, y), rk))

λ (B((x, y), rk))= 1 ∀ (x, y) ∈ E ,

which implies, by (4.95), (4.102) and then (4.101). Since it is trivial that the sequenceof sets

Ek(x, y) := B((x, y), rk) k = 1, 2, . . . , (x, y) ∈ R2

is a differentiation basis for λ, by applying Theorem 2.23 and (4.100)

∃ 1 = limk→∞

µ(Ek(x, y))

λ(Ek(x, y))= Dλµ(x, y) λ-a.e. (x, y) ∈ spt(λ) ,

which implies (4.94).


The notion of rectifiable set in a metric space was already introduced by Federer[Fe, 3.2.14].

Definition 4.29. A set Γ ⊂ (X, d) is said to be countably Hk-rectifiable if

Γ = Γ0 ∪ (∪∞i=1fi(Ai)) ,

where Hk(Γ0) = 0, Ai ⊂ Rk are Lk-measurable and fi : Ai ⊂ (Rk, ‖ · ‖Rk) → (X, d)(i = 1, 2, . . . ,) are Lipschitz functions.

Definition 4.30. A metric space (X, d) is said to be purely Hk- unrectifiable if foreach Lipschitz function f : A ⊂ (Rk, ‖ · ‖Rk)→ (X, d),

Hk(f(A)) = 0 .

A quite general structure rectifiability result can be obatined for theH1-rectifiability(see, for instance, [AT, Theorem 4.4.8]).

Theorem 4.31. If (X, d) is complete, Γ ⊂ X is closed and connected, and H1(Γ) <∞, then there exist countably many Lipschitz curves γh : [0, 1] → Γ (h = 1, 2, . . . )such that

H1 (Γ \ ∪∞h=1γh([0, 1])) = 0 .

In particular Γ is countably H1-rectifiable.

The study of higher Hk-rectifiability with k > 2 in a metric space is much harder.A systematic study of rectifiable sets in general metric spaces was made by Ambrosioand Kirchheim [AK] in 2000. However, the definitions they used are not alwaysappropriate in some remarkable class of metric spaces such as the one called Carnotgroups or also sub-Riemannian stratified groups. Indeed Ambrosio and Kirchheimproved the following result.

Theorem 4.32. ([AK, Theorem 7.2]) The first Heisenberg group (H1, d) is purelyk-unrectifiable for k = 2, 3, 4, for each invariant distance d.

Therefore, taking the previous unrectifiability results into account, a new suitablenotion of rectifiability in Carnot groups is needed, better fitting the new geometry.This study is still object of the current research and an account can be found in [SC2].

132

5. An introduction to minimal surfaces and sets of finite perimeter.([AFP, G, G2, Mag, MM])

Motivation: An introduction to the so-called Plateau’s problem for non-parametricminimal surfaces as far as the problem of existence, uniqueness and regularity isconcerned. An introduction to the sets of finite perimeter and their relationshipswith the minimal surfaces.

5.1. Plateau problem: nonparametric minimal surfaces in Rn, area func-tional and its minimizers. Here we are going to deal with the problem of leastarea for the so-called non parametric hypersurfaces in Rn, that is hypersurfaces whichare graphs of functions. More precisely we will consider an hypersurface S ⊂ Rn with

S = Su := (z, u(z)) : z ∈ ωwhere u ∈ C1(ω) and ω ⊂ Rn−1 is a bounded open set with smooth boundary. Nonparametric minimal surfaces are a particular case of the general theory of (parametric)minimal surfaces, where also surfaces satisfying Definition 4.1 are allowd and theymay be not graphs. A recent account of the development and open problems of thistheory can be found in [Pe].

By (3.74), the (n− 1)-dimensional area of graph Su is given by

(5.1) A (u) := Hn−1(Su) =

∫ω

√1 + |∇u|2 dx

and this formula holds true for each funnction u ∈ Lip(ω). Thus we can define theclassical area functional

A ≡ A (·, ω) : Lip(ω)→ [0,∞), A (u) = A (u, ω) :=

∫ω

√1 + |∇u|2 dx .

This gives rise to the classical Plateau’s problem, that is to show the existence ofan area minimizing hypersurface with a given boundary. More precisely, if we fixa boundary datum g : ∂ω → R we are concerned with the geometric variationalproblem of the calculus of variations, also called Dirichlet problem,

(PP) min A (u, ω) : u ∈ Lip(ω, g) ,where Lip(ω, g) denotes the class of competitors functions

Lip(ω, g) := u ∈ Lip(ω) : u = g on ∂ω ,which is supposed to be nonempty. As usual in the calculus of variations, the maintwo questions about (PP) concerns:

(EUPP) existence and uniqueness of a minimizer u0 of (PP), that is whether there isa function u0 ∈ Lip(ω, g) satisfying

A (u0) 6 A (u) ∀u ∈ Lip(ω, g)

and whether u0 is unique;(RPP) regularity of minimizers, that is whether a minimizer is (locally) regular in ω,

or also (globally) regular in ω provided that both datum g and boundary ∂ωare regular.

133

Here we will mainly deal with the existence of minimizers, and we will only mentionlater some regularity results as far as minimal boundaries, which applies to problem(PP), too.

Existence problem for (PP) essentially was studied by means of two strategies (see[G2, Introduction] for a more complete and interesting account of this issue).

• The first strategy is by studying the associated Euler-Lagrange equation toarea functional A . Namely

Exercise: prove that, given ϕ ∈ C∞(ω) and u0 minimizer of (PP), let R 3t 7→ a(t) := A (u0 + tϕ), then t = 0 is a minimum point of a and

∃ 0 = a′(0) =

∫ω

∇u · ∇ϕ√1 + |∇u|2

dz .

The previous identity yields the associated Euler-Lagrange equation to areafunctional A , called minimal surface equation and, in divergence form, it readsas

(MSE) div

(∇u√

1 + |∇u|2

)= 0 in ω .

(MSE) is a nonlinear PDE equation, which has been deeply studied for twocenturies and half long with regard to existence, uniqueness and regularity ofits solutions. Then one transfers those results about solutions of (MSE) tominimizers of area functional (see [G, G2, MM]).• The second strategy consists in showing directly the existence of minimizers

for the area functional without studying its associated Euler-Lagrange, thatis the so-called direct methods of the calculus of variations. In the followingwe will follow this strategy first for problem (PP) and then for the generalizedproblem (PP) dealing with sets of finite perimeter.

Historical notes: Plateau’s problem originates much before with Lagrange whoin 1762 studied it, derived the minimal surface equation (MSE) and gives a foundationof the calculus of variations. We have also to acknoweledge that Euler in 1741 alreadyexibited an example of minimal surface by means of the catenoid (see Example 5.6).The historical development of the theory of minimal surfaces involves many of thegreatest mathematicians of their time and an account can be found in [Pe].

5.2. Direct method of the calculus of variations and application to theexistence of minimizers for the Plateau problem. One of the general resultsapplying the direct methods in calculus of variations is the following generalizedWeierstrass theorem for the existence of minimizers.

Theorem 5.1 (Generalized Weierstrass theorem). Let (X, τ) be a topological spaceand let F : X → (−∞,+∞] be a functional such that

(i) F is sequentially lower semicontinuous (slsc), i.e. for each x ∈ X and sequence(xh)h ⊂ X with limh→∞ xh = x, then

F (x) 6 lim infh→∞

F (xh) ;

134

(i) F is sequentially coercive, that is for each t ∈ R there exists a sequentiallycompact set Kt ⊂ X such that

x ∈ X : F (x) 6 t ⊆ Kt .

Then ∃minX F .

Proof. If F ≡ +∞, the conclusion is trivial, otherwise let (xh)h ⊂ X be such that

limh→∞

F (xh) = m := infXF < +∞ .

If t > m, there exist h and a sequentially compact set Kt such that

xh ∈ x ∈ X : F (x) 6 t ⊆ Kt ∀h > h .

Up to a subsequence, we can suppose, by claim (ii), that there exist x ∈ Kt such thatlimh→∞ xh = x. Let us show

F (x) = m,

which will imply our conclusion. Indeed, by claim (i),

F (x) 6 lim infh→∞

F (xh) = limh→∞

F (xh) = m = infXF ,

and we reach the desired conclusion.

A classical application of the direct methods of the calculus of variations to problem(PP) can be given by the so-called bounded slope condition property for boundarydatum g, which goes back to Hilbert, and we recall here.

Definition 5.2. We say that a function g : ∂ω → R satisfies the bounded slopecondition with constant Q > 0 (Q-B.S.C. for short, or simply B.S.C. when the constantQ does not play any role) if for every z0 ∈ ∂ω there exist two affine functions w+

z0and

w−z0 such that

w−z0(z) 6 g(z) 6 w+z0

(z) ∀z ∈ ∂ω,w−z0(z0) = g(z0) = w+

z0(z0)

Lip(w−z0) 6 Q and Lip(w+z0

) 6 Q,

where Lip(w) denotes the Lipschitz constant of w.

We also recall that a set ω ⊂ Rn−1 is said to be uniformly convex if there exist apositive constant C = C(ω) and, for each z0 ∈ ∂ω, a hyperplane Πz0 passing throughz0 such that

|z − z0|2 6 C d(z,Πz0) ∀z ∈ ∂ω,where d(z,Πz0) := inf|z − w| | w ∈ Πz0.

Remark 5.3. We collect here some facts on the B.S.C.

a) If g : ∂ω → R satisfies the B.S.C. and is not affine, then ω has to be convex(see [G2, page 20]) and g is Lipschitz continuous on ∂ω. Moreover, if ∂ω hasflat faces, then g has to be affine on them.

This property seems to say that the B.S.C. is a quite restrictive assumption. Anyhowthe following one, due to M. Miranda [Mi] (see also [G2, Theorem 1.1]), shows thatthe class of functions satisfying the B.S.C. on a uniformly convex set is quite large.

135

b) Let ω ⊂ Rn be open, bounded and uniformly convex; then every g ∈ C1,1(Rn)satisfies the B.S.C. on ∂ω.

Theorem 5.4. Let ω be a bounded open set in Rn−1 and assume that g : ∂ω → Rsatisfies B.S.C. with constant Q > 0. Then problem (PP) has a unique minimizeru0 ∈ Lip(ω, g). Moreover u0 satisfies the estimate

Lip(u0) 6 Q .

Proof. See, for instance, [G2, Theorem 1.2].

Remark 5.5. In Theorem 5.4 we obtained existence and uniqueness of minimizersunder special assumptions of convexity on the domain ω. If we want to weaken theseconditions, so as to treat more general domains ω, we could use new comparisonfunctions, more general than the affine functions of the B.S.C. Indeed it is possible toprove, by means of the barriers method, the existence and uniqueness of a minimizerfor (PP) in Lip(ω, g), provided that bounded open set ω has a boundary of class C2,with non-negative mean curvature and g is of class C2 (see [G2, Theorem 1.6]). Letus recall that if ω is an open bounded set of class C2 and convex, then its boundaryhas non-negative mean curvature (see [G2, Page 29]). The condition on non-negativemean curvature on the boundary is almost necessary. In fact, one can prove that ifthe mean curvature is negative at some point of boundary ∂ω, then there exists aregular function g for which the area functional has no minimum in Lip(ω, g) (see[G2, Theorem 1.7]). We will show by means of an example such an eventuality.

Example 5.6 (Non-existence for Plateau’s problem). Let n = 3 and let ω ⊂ R2 bethe annulus

(5.2) ω := z = (x, y) ∈ R2 : % < |z| < R

with 0 < % < R given. Consider problem (PP) with boundary datum

(5.3) g(z) =

0 if |z| = RM if |z| = % ,

with M > 0. We will show that this problem admits no minimizer when M is largeenough.

We begin by proving that, if a minimizer exists, then there exists a rotationallyinvariant one. To this aim, it is enough to prove that for any u ∈ Lip(ω) we have

(5.4)

∫ω

√1 + |∇u|2 dL2 6

∫ω

√1 + |∇u|2 dL2

where, after setting Rθ to be the rotation in R2 of an angle θ, that is the linearisometry Rθ : R2 → R2

Rθ(x, y) := (cos θ x− sin θ y, sin θ x+ cos θ y) ,

we define the rotationally symmetric function u : ω → R by

(5.5) u(z) :=

∫ 2π

0

(u Rθ)(z) dθ =

∫ 2π

0

u(|z| cos θ, |z| sin θ) dθ .

Indeed, when u ∈ Lip(ω) one has

136

Exercise:

(5.6) ∇(u Rθ) = R−θ (∇u) Rθ L2-a.e. in ω .

By (5.6) and since Rθ and R−θ are isometries, it follows that, for L2-a.e. z ∈ ω,

|∇u(z)| =∣∣∣∣ ∫ 2π

0

∇(u Rθ)(z) dθ

∣∣∣∣=

∣∣∣∣ ∫ 2π

0

(R−θ ∇u Rθ) (z) dθ

∣∣∣∣6∫ 2π

0

|R−θ ∇u Rθ| (z) dθ

=

∫ 2π

0

|∇u| (z) dθ = |∇u(z)| .

(5.7)

Thus, by (5.7), (5.4) is proved . Moreover, it is not difficult to show that

(5.8) u|∂ω = g

for any u ∈ Lip(ω, g), that is that also u ∈ Lip(ω, g).We are now going to exclude the existence of rotationally invariant minimizers.

Let u(z) = u(x, y) = v(√

x2 + y2)

where

v ∈ Lip∗(%,R) := v ∈ Lip([%,R]) : v(%) = M and v(R) = 0 .Then it is easy to set that u ∈ Lip(ω, g) and

|∇u(z)| = |v′ (|z|)| L2-a.e. z ∈ ω .Thus, we obtain

(5.9)

∫ω

√1 + |∇u|2 dL2 = 2π

∫ R

%

r√

1 + v′(r)2 dr =: L(v) .

We are going to show that for M 1 the functional L does not admit minimizersin the one-dimensional class Lip∗(%,R).

Suppose now that v is a minimizer in Lip∗(%,R).

Exercise: Prove that function

[%,R] 3 r 7→ r v′(r)

1 + v′(r)2is absolutely continuous

and

(5.10)d

dr

(r v′(r)

1 + v′(r)2

)= 0 L1-a.e. r ∈ [%,R] .

(Hint: Let us consider the Euler-Lagrange equation associated to L: let ϕ ∈C1c((%,R)) be given and let l(t) := L(v+ t ϕ) if t ∈ R; then l′(0) = 0 and deduce the

desired conclusion.)By (5.10), it follows that

(5.11)r v′(r)√1 + v′(r)2

= c ∀ r ∈ [%,R] ,

137

for a suitable c ∈ R. In particular for r ∈ [%,R]

|c| =

∣∣∣∣∣ r v′(r)√1 + v′(r)2

∣∣∣∣∣ 6 |r|and so |c| 6 %. From (5.11) and taking into account sgn v′ = sgn c we obtain

(5.12) v′(r) =c√

r2 − c2=

sgn c√(r/c)2 − 1

L1-a.e. r ∈ [%,R] .

By (5.12), v has to be monotone, and since v(%) = M > v(R) = 0, v is not increasing.Thus C < 0, and, without loss of generality, by replacing c with −c,the solution isgiven by

v(r) =

∫ R

r

dσ√(σ/c)2 − 1

=

∫ R/c

r/c

c dσ√σ2 − 1

= c (arccosh(R/c)− arccosh(r/c)) if % 6 r 6 R .

(5.13)

where, by definition, arccosh : [1,∞) → [0,∞) is the inverse function of hyperboliccosine function

cosh : [0,∞)→ [1,∞), cosh(t) :=et + e−t

2and it can be explicitely represented as

arccosh(s) = log(s+√s2 − 1

)if s > 1 .

Moreover the constant 0 < c 6 % has to be chosen in order that v(%) = M . Observethat we can equivalently represent v as

(5.14) v(r) = c logR +√R2 − c2

r +√r2 − c2

if % 6 r 6 R ,

and

(5.15) M = v(%) = c logR +√R2 − c2

%+√%2 − c2

6 supc∈(0,%]

(c log

R +√R2 − c2

%+√%2 − c2

).


(5.16) supc∈(0,%]

(c log

R +√R2 − c2

%+√%2 − c2

)= % log

R +√R2 − %2

%=: M0(%,R) .

(Hint: Prove that the function (0, %] 3 c 7→ c logR +√R2 − c2

%+√%2 − c2

is nondecreasing.)

By (5.14), (5.15) and (5.16), it follows that problem (PP) can be solved onlyif M < M0, when ω is the open set in (5.2) and g is the boundary datum in (5.3) .In the limit case M = M0 we have c = % and,

|∇u(x, y)| =∣∣∣v′ (√x2 + y2

)∣∣∣ =%√

x2 + y2 − %2

138

becomes infinite on the internal circumference and then u is not admissible becauseu /∈ Lip(ω). However a minimizer exists in a larger class of competitors, namely inthe Sobolev space W 1,1(ω), as proved in the following exercise.

Exercise: Let us define the space of functions

AC∗(%,R) := v ∈ AC([%,R]) : v(%) = M0 and v(R) = 0 .(i) Prove that functional L in (5.9) is well-defined on AC∗(%,R), that is, L :

AC∗(%,R)→ [0,∞).(ii) Let v0 be the function in (5.14) with c = %. Prove that v0 is a minimizer of

functional L : AC∗(%,R)→ [0,∞), that is, v0 ∈ AC∗(%,R) and

L(v0) 6 L(v) ∀ v ∈ AC∗(%,R) .

(iii) Let

(5.17) uh(z) := vh(|z|), u0(z) := v0(|z|) if z ∈ ω .where

vh(r) := ch % logR +

√R2 − %2 + 1/h

r +√r2 − %2 + 1/h

if % 6 r 6 R

and the constant ch is chosen in such a way that

vh(%) = M0 .

Prove that

(5.18) (uh)h ⊂ C1(ω) ∩ Lip(ω, g),

(5.19) uh → u0 uniformly on ω, ∇uh → w in (L1(ω))2

where

w(z) := v′0(|z|) z|z|

if % < |z| < R .

Moreover

(5.20) L(v0) = limh→∞

L(vh) = limh→∞

A (uh, ω) = inf A (u, ω) : u ∈ Lip(ω, g) ,

where g is the boundary datum in (5.3) with M = M0.(iv) Prove that

u0 ∈ W 1,1(ω) ∩C0(ω) ,

that is, by definition, u0 ∈ C0(ω), u0 ∈ L1(ω) and there exists its weak gradient

Du0(z) = w(z) if % < |z| < R ,

with Du0 ∈ (L1(ω))2. Moreover u0 /∈ C1(ω) as well u0 /∈ Lip(ω), even if

L(v0) =

∫ω

√1 + |Du0|2 dx .

If M > M0 there is no solution to problem (PP). In this case, one canprove that, by means of the direct methods and by replacing the space of Lipschitzcontinuous functions with the one of bounded variation functions , the minimal surfaceis given by the graph of the solution u corresponding to the limit value M0, plus the

139

portion of the vertical cylinder having for base the internal circumference of radius%, that lies between the levels M0 and M (see [G, Chapter 14]).

Finally let us notice that, in the case M < M0, the function v in (5.13) is theinverse function of

r : [0,M ]→ [%,R], r(v) := c cosh

(v − bc

)where b := c arccosh(R/c), which is an arc of catenary joining the points (0, R) and(M,%) in the plane v, r. By rotation about the v-axis, it generates a surface calledcatenoid, graph of u, which is a surface of revolution of minimal area. Surfaces ofrevolution of minimal area was an issue very studied in the history of calculus ofvariations, starting from Euler. An account of this fascinating development can befound in [GH, Chap. 5, Sect. 2.4, Example 5].

Historical notes: [MM] Variational problems concerning manifolds, one or moredimensional, immersed in an Euclidean space are among the most classical ones. Wemean that they have been considered since Bernoulli’s time and have not obtaineda general satisfactory treatment until the the 1950s. At the start of the last centuryvery interesting new ideas about variational problems for surfaces, are contained in thethesis of Lebesgue ”Integrale, Longueur, Aire”. Of the same period of time, are theinteresting papers of Tonelli about the length of the curves. In the 1930s appeared therelevant series of papers by Douglas and Rado about the Plateau Problem, togetherwith some interesting contributions of Tonelli concerning variational problems withtwo independent variables. The school of Tonelli, particularly Cesari, worked at theproblem of a definition of the surface area, good from the variational point of view.But it was only in the 1950s that new definitions of surfaces were introduced andused for a general treatment of classical variational problems like the isoperimetricproperty of the sphere and the Plateau Problem. In the new approaches, like Reifen-berg surfaces, Federer-Fleming integral currents, De Giorgi perimeters and Almgrenvarifolds, ideas from the Modern Algebra, General Measure Theory and DistributionTheory are used together with the classical arguments from Differential Geometryand Real Variable Functions Theory.

5.3. Sets of finite perimeter, space of bounded variation functions and theirmain properties; sets of minimal boundary. We are going to introduce analternative approach to the theory minimal surfaces, which applies to hypersurfacesof Rn, that is surfaces of topological dimension n−1 and which extend the one studiedbefore for non-parametric minimal surfaces. The main idea is that a hypersurface inRn is meant as boundary of a set E ⊂ Rn whose characteristic function χE has boundedvariation, namely E is a set of finite perimeter. The notion goes back to De Giorgi,who introduced it in the pioneering papers [DG1, DG2], strongly inspired by someprevious ideas of Caccioppoli (see [A2] for an interesting account of Euclidean setsof finite perimeter). Caccioppoli’s primitive idea, then refined by De Giorgi throughsets of finite perimeter, considered oriented hypersurfaces, which (at least locally)are boundaries of sets, and exploited techniques of measure theory. Let us begin tostress some benefits for the introduction of sets of finite perimeter by means of twoleast-area problems.

140

Problem 1 (Plateau’s problem for general domains). We can give a formulation ofPlateau’ problem (PP) in terms of boundary of a set in Rn+1 as follows. Let ω ⊂ Rn bea bounded open set with regular boundary and let denote with Ω and Ω respectivelythe open and closed cylinders

(5.21) Ω := ω × R, Ω := ω × R .

If v : ω → R, let Ev ⊂ Rn+1 the (closed) subgraph induced by v, that is

(5.22) Ev :=

(z, xn+1) ∈ Ω : xn+1 6 v(z)

and let Sv ⊂ Rn+1 be the graph of v, that is

(5.23) Sv :=

(z, xn+1) ∈ Ω : xn+1 = v(z).

Let u, g : ω → R be Lipschitz continuous functions. Then it is easy to see that

∂Eu ∩ Ω = Su ∩ Ω

and we can mean the boundary condition u = g on ∂ω as

(5.24) Eu = Eg in Rn+1 \ Ω .

Therefore Plateau’s problem (PP) can be formulated and extended in terms of setsas follows

min Hn(∂Eu ∩ Ω) : u ∈ X,Eu satisfies (5.24) where X is a suitable set of functions to be chosen. The benefit of this formulation

is the chance to get existence for more general regular domains ω than the ones strictconvex allowed in Theorem 5.4 and Remark 5.4 (see also Example 5.6). Indeed thisis the case since, by the relaxation method, one can find out that suitable class X ofcompetitors is the the space of functions of bounded variations on ω ( or, equivalently,sets Eu of finite perimeter in cylinder Ω) and then, by a direct methods of the calculusof variation, to get the existence (see [G, Sect. 14.4]).

Problem 2 (Isoperimetric problem). The classical isoperimetric problem is the mostcelebrated problem of least-area and, likely, one of the earliest problem in this argu-ment: it asks to find out a plane figure of the largest possible area whose boundaryhas a specified length. It can be extended to higher dimensions as follows: if E ⊂ Rn

and

|E| := Ln(E), Hn−1(∂E) ,

respectively denote the n-dimensional volume of E and the (n− 1)-dimensional area(we will call also perimeter) of ∂E, to find out a set Eiso with maximum n-volume,among sets E ∈ X with |E| < ∞ and fixed perimeter Hn−1(∂E) = c, where X is asuitable class of admissible sets for the n-volume and perimeter functions, stable bytranslations an dilations, to be chosen (for instance, X could be the class of sets withC1 regular boundary).

It is also well-known that the isoperimetric problem is equivalent to the so-calledisovolumetric problem, that is to find out a set Eiso with minimum perimeter, amongsets E ∈ X with fixed volume |E| = c. Indeed, by means of the isoperimetricinequality, that is the inequality

(II) min|E|(n−1)/n, |Rn \ E|(n−1)/n

6 CHn−1(∂E)

141

which holds for a suitable positive constant C > 0 and for each set E ∈ X, theisoperimetric problem is equivalent to the isovolumetric problem

(IP) m = minHn−1(∂E) : E ∈ X, |E| = 1

since the set functions

X 3 E 7→ |E|(n−1)/n and X 3 E 7→ Hn−1(∂E)

are homogeneous of degree n− 1. Moreover, if problem (IP) has a solution, the valueCiso = 1/m is the minimum constant C for which (II) turns to be true.

It is also well-known that balls are the only solutions of problem (IP) when X isthe class of sets with C1 boundary. Thus we can explicitely calculate the value of Ciso

and it turns out to be

(5.25) Ciso = (nn α∗n)1/(1−n) ,

where α∗n := |B(0, 1)|. Observe that the isoperimetric constant (5.25) could dependon the class X. However De Giorgi [DG3] proved the minimality of balls in the largestadmissible class X, namely the sets of finite perimeter.

An other important feature that class X of competitors in problems 1 and 2 has tosatisfy is the sequential compactness with respect to a suitable topology τ , in order toapply the direct methods of the calculus of variations. It is quite clear that neither theclass of sets with boundary of class C1 nor the one of class Lipschitz are appropriatefor this goal. For instance, let us assume that (Eh)h is the following sequence

Eh = Euh ,

where ω ⊂ R2 is the domain in (5.2) and (uh)h is the sequence of functions definedin (5.17). Then, by the previous exercise, it turns out that (uh)h is a minimizingsequence for Plateau’s problem (PP) in the class Lip(ω, g) and, if Ω := ω × R,

limh→∞H2 (∂Eh ∩ Ω) = lim

h→∞

∫ω

√1 + |∇uh|2 dx =

∫ω

√1 + |Du0|2 dx

but u0 is not neither in C1(ω) nor in Lip(ω), even though (uh)h uniformly convergesto u0.

The fundamental result to take into account for the introduction of sets of finiteperimeter is the Gauss-Green theorem, which holds for sets with regular boundary.

Theorem 5.7 (Gauss-Green). Assume that E ⊂ Rn is an open set with its boundary∂E of class C1 and let g = (g1, . . . , gn) ∈ C1

c(Rn,Rn) ≡ (C1c(Rn))n. Then

(GG)

∫E

div(g) dLn =

∫∂E

(NE, g)Rn dHn−1 ,

where NE : ∂E → Sn−1 denotes the (continuous) outward unit normal to E and

div(g)(x) :=n∑i=1

∂ gi∂xi

(x) x ∈ Rn .

Proof. See, for instance, [Mag, Theorem 9.3].

142

Remark 5.8. Notice that if E is an open set with C1 regular boundary ∂E, then itsoutward normal boundary NE : ∂E → Sn−1 can be extended to a continuous functionNE : Rn → Rn satisfying |NE(x)| 6 1 for each x ∈ Rn (see [Mag, 9.2]).

Gauss-Green formula (GG) can be read in the sense of distributions by means ofmeasure theory as follows:

(5.26)

∫RnχE div(g) dLn = −

∫Rn

(−NE, g)Rn d(Hn−1 ∂E) ∀ g ∈ C1c(Rn,Rn) .

By (5.26) and measure theory, we can infer the following suggestions in order to definethe class of sets of finite perimeter:

• let g = ϕ ei with ϕ ∈ C1c(Rn), where e1, . . . , en denotes the standard basis

of Rn, then, by (5.26) it follows that, if NE = (N(1)E , . . . , N

(n)E ),∫

RnχE ∂iϕdLn = −

∫Rnϕ (−N (i)

E ) d(Hn−1 ∂E) ∀ϕ ∈ C1c(Rn), i = 1, . . . , n ,

that is, characteristic function χE admits a weak gradient DχE in Rn repre-sented by finite Radon vector measure νE = −NEHn−1 ∂E : B(Rn)→ Rn;• by (5.26) it follows that

(5.27)

∫RnχE div(g) dLn 6 Hn−1(∂E) ‖g‖∞ ∀ g ∈ C1

c(Rn,Rn) ,

where ‖g‖∞ := supRn |g|. In particular, by the previous inequality and thedensity of C1

c(Rn,Rn) ≡ (C1c(Rn))n in ((C0

c(Rn))n, ‖ · ‖∞), it follows that thelinear functional LE : C1

c(Rn,Rn) ≡ (C1c(Rn))n → R

LE(g) :=

∫RnχE div(g) dLn

can be extended to a linear functional LE : (C0c(Rn))n → R, which is also

continuous according to Definition 1.69.

Taking the previous suggestions into account, we can introduce the space of func-tions with bounded variation on Rn, which is a bigger class of sets of finite perimeter.

Definition 5.9. Let Ω ⊆ Rn be an open set.

(i) If u ∈ L1(Ω) we call variation of u on Ω the value

(5.28) |Du|(Ω) := sup

∫Ω

u(x)divg dLn : g ∈ C1c(Ω,Rn), |g(x)| 6 1

∈ [0,∞].

(ii) We say that u ∈ L1(Ω) has bounded variation in Ω if |Du|(Ω) < ∞.

The space BV (Ω) is the set of functions u ∈ L1(Ω) with bounded variation in Ω. Thespace BVloc(Ω) is the set of functions u : Ω → R such that u|Ω′ ∈ BV (Ω′) for eachopen set Ω′ b Ω.

Exercise: Let Ω ⊂ Rn be an open set and let u ∈ C1(Ω), Then

|Du|(Ω) =

∫Ω

|∇u| dx .

143

(Hint: By (GG)∫Ω

u(x)divg dLn =

∫Ω

(g,∇u)Rn dLn ∀ g ∈ C1c(Ω,Rn)) .

Theorem 5.10 (Structure of BV functions). Let Ω ⊂ Rn be an open set and letu ∈ BVloc(Ω). Then there exist a unique Radon measure µu : B(Ω) → [0,∞] and a

Borel measurable vector function wu = (w(1)u , . . . , w

(n)u ) : Ω→ Sn−1 such that

(5.29) µu(A) = |Du|(A) for each open set A ⊂ Ω,

(5.30)

∫Ω

u divg dLn =

∫Ω

(wu, g)Rn dµu,

for all g ∈ C1c(Ω,Rn). Moreover

Du = (D1, . . . , Dn) := −(w(1)u , . . . , w(n)

u )µu : Bcomp(Ω)→ Rn

is a Radon vector measure such that

(5.31)

∫Ω

u ∂iϕdLn =

∫Ω

ϕw(i)u dµu := −

∫Ω

ϕdDiu ∀ϕ ∈ C1c(Ω), i = 1, . . . , , n .

Viceversa if (5.31) holds for some u ∈ L1(Ω), a Radon measure µ ≡ µu and functions

wi ≡ w(i)u ∈ L1

loc(Ω, µ) (i = 1, . . . , n), then u ∈ BVloc(Ω) and (5.29) and (5.30)respectively holds with

(5.32) µu(B) :=

∫B

√√√√ n∑i=1

w2i dµ if B ∈ B(Ω) ,

(5.33) wu(x) :=

(w1, . . . , wn)√∑n

i=1w2i

(x) if 0 <∑n

i=1 w2i (x) < ∞

0 otherwise

.

Notation: In the following, if u ∈ BVloc(Ω), since (5.29), we will identify variation|Du| (5.28) with measure µu.

Proof. 1st step: Let us prove there exist a unique Radon measure µu : B(Ω)→ [0,∞]and a Borel vector function wu : Ω→ Sn−1 such that (5.29) and (5.30) hold.

Let Lu : C1c(Rn,Rn) ≡ (C1

c(Rn))n → R be the linear functional

Lu(g) :=

∫Rnu div(g) dLn .

Arguing as in (5.27), it follows that, for each open set Ω′ b Ω,

(5.34) |Lu(g)| 6 |Du|(Ω′) ‖g‖∞ ∀ g ∈ C1c(Ω

′,Rn) .

Let K ⊂ Ω be a compact. Then there exists an open set Ω′ such that

K ⊂ Ω′ b Ω .

Let g ∈ (C0c(Ω))n with spt(g) ⊂ K and let

(5.35) gh(x) := ((g1 ∗ %h)(x), . . . , (gn ∗ %h)(x)) if x ∈ Ω ,

144

where (%h)h is a sequence of mollifiers in Rn. Then, it is easy to see that

gh ∈ C1c(Ω

′,Rn) if1

h< d(K, ∂Ω′) ,

|gh(x)| 6 ‖g‖∞ ∀x ∈ Ω′, h ,

limh→∞‖gh − g‖∞ = 0 .

By (5.34), sequence (L(gh))h ⊂ R is a Cauchy sequence. Thus we can extendfunctional Lu to a linear functional Lu : (C0

c(Ω))n → R as follows

Lu(g) := limh→∞

Lu(gh)

and the limit is independent of the choice of the sequence (gh)h converging to g.Moreover

supLu(g) : g ∈ (C0

c(Ω))n, |g| 6 1, spt(g) ⊂ K6 |Du|(Ω′) < ∞ .

Therefore linear functional Lu : (C0c(Ω))n → R is continuous according to Definition

1.69. Applying the Riesz representation theorem 1.73 and defining

µu := µLu , wu := wLu ,

(5.29) and (5.30) follow.2nd step: To prove (5.31), it is sufficient to choose g := ϕ ei (i = 1, . . . , n) and

use (5.30).3rd step: Suppose that (5.31) holds for some u ∈ L1(Ω), a Radon measure

µ ≡ µu and functions wi ≡ w(i)u ∈ L1

loc(Ω, µ) (i = 1, . . . , n), then let us prove that u ∈BVloc(Ω) and (5.29) and (5.30) respectively hold with µu and wu given by, respectively,(5.32) and (5.33). By assumptions, for each g = (g1, . . . , gn) ∈ C1

c(Ω,Rn), if w =(w1, . . . , wn), ∫

Ω

u divg dLn =

∫Ω

un∑i=1

∂igi dLn =n∑i=1

∫Ω

u ∂igi dLn

=n∑i=1

∫Ω

wi gi dµ =

∫Ω

n∑i=1

wi gi dµ

=

∫Ω

(w, g)Rn dµ .

(5.36)

Let us fix an open set Ω′ b Ω and let g ∈ C1c(Ω

′,Rn). Since w ∈ (L1loc(Ω))n, it follows

that

(5.37)

∫Ω

(w, g)Rn dµ 6 ‖g‖∞∫

Ω′|w| dµ <∞ .

Thus, by (5.36) and (5.37), we obtain that

|Du|(Ω′) 6∫

Ω′|w| dµ <∞ ,

for each open set Ω′ b Ω, that is, u ∈ BVloc(Ω). By (5.30) and (5.36), it follows that

(5.38)

∫Ω

(wu, g)Rn dµu =

∫Ω

(w, g)Rn dµ ,

145

for each g ∈ C1c(Ω,Rn). By using the approximation by convolution defined in (5.35),

one can prove that (5.38) still holds for each g ∈ C0c(Ω,Rn). The approximation by

continuous functions in Lp (see Remark 1.64) enables us to use functions g = wu χΩ′

and g =w

|w|χΩ′ in (5.38), for each open set Ω′ ⊂ Ω, which yields inequalities

µu(Ω′) 6

∫Ω′|w| dµ and µu(Ω

′) >∫

Ω′|w| dµ ,

that is

µu(Ω′) =

∫Ω′|w| dµ for each open set Ω′ ⊂ Ω ,

Thus (5.32) follows by the approximation in measure of Borel sets by means ofopen sets from above. By (5.32) and (5.38), (5.33) follows.

BV functions of one variable ([AFP, Sect. 3.2]) ????.

Definition 5.11. A (Lebesgue) measurable set E ⊂ Rn is of locally finite perimeterin an open set Ω ⊂ Rn (or is a Caccioppoli set) if the characteristic function χE ∈BVloc(Ω). In this case we call the perimeter of E the measure

(5.39) |∂E| := |DχE|and we call the (generalized inward unit) normal to ∂E in Ω the vector

(5.40) νE(x) := −wχE(x).

Remark 5.12. Observe that, by (5.40) and (5.30), it follows that, if E ⊂ Rn is aset of locally finite perimeter in an open set Ω ⊂ Rn, then it satisfies, in sense ofdistributions, the following generalized Gauss-Green formula

(5.41)

∫E

div(g) dLn = −∫

Ω

(νE, g)Rn d|∂E| ∀ g ∈ C1c(Ω,Rn) .

Thus vector function νE : Ω → Sn−1 acts as the inward normal to E in the classicalGauss-Green formula (see (5.26)).

Theorem 5.13. If E ⊂ Rn is an open set with C1 boundary , then E is a set oflocally finite perimeter in Rn and for each open set Ω ⊂ Rn

(5.42) |∂E|(Ω) = Hn−1(∂E ∩ Ω) .

Proof. By definition of perimeter and (GG)

|∂E|(Ω) := sup

∫E

divg dLn : g ∈ C1c(Ω,Rn), |g| 6 1

= sup

∫∂E∩Ω

(NE, g)Rn dHn−1 : g ∈ C1c(Ω,Rn), |g| 6 1

6 Hn−1(∂E ∩ Ω)

(5.43)

where NE : ∂E → Sn−1 is the outward unit normal to E. To prove the reverseinequality, let us observe that, by Remark 5.8, there exists a continuous functionNE : Rn → Rn with

(5.44) NE|∂E = NE and |NE(x)| 6 1 ∀x ∈ Rn .

146

Let ϕ ∈ C1c(Ω) with 0 6 ϕ 6 1 and let

gh(x) :=(

(ϕNE) ∗ %h)

(x) x ∈ Rn ,

where (%h)h is a sequence of mollifiers in Rn and gh is the convolution product in(5.35). It is easy to see that, by (5.44),

gh ∈ C1c(Ω,Rn) for h large ,

andgh → ϕNE uniformly on Rn, |gh(x)| 6 ϕ(x) ∀x ∈ Rn .

Therefore, for given ϕ ∈ C1c(Ω) with 0 6 ϕ 6 1, by the dominated convergence

theorem,

sup

∫∂E∩Ω

(NE, g)Rn dHn−1 : g ∈ C1c(Ω,Rn), |g(x)| 6 1

> lim

h→∞

∫∂E∩Ω

(NE, gh)Rn dHn−1 =

∫∂E∩Ω

ϕdHn−1 .

(5.45)

By (5.43) and (5.45), taking the supremum on all ϕ ∈ C1c(Ω) with 0 6 ϕ 6 1, it

follows that|∂E|(Ω) > Hn−1(∂E ∩ Ω) .

Remark 5.14. Notice that (5.42) may not hold true if E is a set of finite perimeterbut its boundary ∂E is no more regular. Indeed there exists a set E ⊂ Rn (n > 2)of finite perimeter in Rn, that is |∂E|(Rn) < ∞, but 0 < Hn(∂E) = Ln(∂E) < ∞(see, for instance,[Mag, Example 12.25]). In particular Hn−1(∂E) = ∞ . Thus a setof finite perimeter may have a wild topological boundary.

Remark 5.15. The perimeter is invariant under translations, that is

(5.46) |∂(p+ E)|(p+ A) = |∂E|(A), ∀p ∈ Rn and for any Borel set A ⊂ Rn ,

ifp+ E := p+ x : x ∈ E .

Indeed the differential operator div is invariant under translations and the n-dimensionalLebesgue measure Ln is invariant under traslations. Moreover the perimeter is ho-mogeneous of degree n− 1 with respect to the dilations , that is

(5.47) |∂(λE)|(λA) = λn−1|∂E|(A), ∀λ > 0 and for any Borel set A ⊂ Rn ,

ifλE := λx : x ∈ E .

Also this fact is elementary and can be proved by changing variables in formula(5.28).

Let us recall some simple properties of sets of (locally) finite perimeter,

Proposition 5.16. L Ω ⊂ Rn be an open set and let E and F be measurable subsetsof Rn. Then

(i) spt(|∂E|) ⊂ ∂E;(ii) |∂E|(Ω) = |∂(Rn \ E)|(Ω);

147

(iii) (locality of the perimeter measure) |∂E|(Ω) = |∂(E ∩ Ω)|(Ω);(iv) |∂(E ∪ F )|(Ω) + |∂(E ∩ F )|(Ω) 6 |∂E|(Ω) + |∂F |(Ω).

Proof. See [AFP, Proposition 3.38].

The direct methods of the calculus of variations apply to the sets of finite perimeterand, more generally, to the space of bounded variations since they satisfies the twoimportant properties of semicontinuity and compactness.

Theorem 5.17 (Lower semicontinuity in BV). Let u, uh ∈ L1loc(Ω), h ∈ N and

suppose that uh → u in L1loc(Ω), that is

uh → u in L1(K) as h→∞, for each compact set K ⊂ Ω .

(i) Thenlim infh→∞

|Duh|(Ω) > |Du|(Ω) .

(ii) Assume also that

sup |Duh|(Ω′) : h ∈ N < ∞ for each open set Ω′ b Ω .

Then u ∈ BVloc(Ω) and Duh∗Du in Ω, that is,∫

Ω

ϕdDuh →∫

Ω

ϕdDu ∀ϕ ∈ C0c(Ω) .

Proof. See [AFP, Propositions 3.6 and 3.16].

Theorem 5.18 (Compactness in BV ). Every sequence (uh)h ⊂ BVloc(Ω) satisfying

(5.48) sup

∫Ω′|uh|dx+ |Duh|(Ω′) : h ∈ N

< ∞ for each open set Ω′ b Ω ,

admits a subsequence (uhk)k converging in L1loc(Ω) to u ∈ BVloc(Ω). If Ω is a bounded

open set with C1 boundary, (uh)h ⊂ BV (Ω) and (5.48) holds with Ω′ ≡ Ω, thenu ∈ BV (Ω) and

uhk → u in L1(Ω) as k →∞ .

Proof. See [AFP, Theorem 3.23].

The previous results for the space of bounded variations simply yield the followingones for sets of finite perimeter.

Definition 5.19. Given (Lebesgue) measurable sets (Eh)h and E in Rn and an openset Ω ⊂ Rn,

(i) we say that (Eh)h locally converges to E in Ω, and write Ehloc→E in Ω, if

χEh → χE in L1loc(Ω), which amounts to

|K ∩ (Eh∆E)| → 0 as h→∞, for each compact K ⊂ Ω .

(ii) We say that (Eh)h converges to E in Ω, and write Eh → E in Ω, if χEh → χEin L1(Ω), which amounts to

|Ω ∩ (Eh∆E)| → 0 as h→∞ .

Corollary 5.20 (Lower semicontinuity for sets of finite perimeter). Let (Eh)h and E

be measurable sets of Rnand suppose Ehloc→E.

148

(i) Thenlim infh→∞

|∂Eh|(Ω)) > |∂E|(Ω)) .

(ii) Assume also that

sup |∂Eh|(Ω′) : h ∈ N < ∞ for each open set Ω′ b Ω .

Then E is a set of locally finite perimeter in Ω and νEh |∂Eh|∗νE |∂E| in Ω,

that is, ∫Ω

ϕνEh d|∂Eh| →∫

Ω

ϕνE d|∂E| ∀ϕ ∈ C0c(Ω) .

where νF denotes the generalized inward normal to F , if F is a set of locally finiteperimeter.

Proof. The proof is immediate by Theorem 5.17 .

Corollary 5.21 (Compactness for sets of finite perimeter). Let (Eh)h be a sequenceof sets with locally finite perimeter in an open set Ω ⊂ Rn, satisfying

(5.49) sup |Ω′ ∩ Eh|+ |∂Eh|(Ω′) : h ∈ N < ∞ for each open set Ω′ b Ω .

Then there exist a subsequence (Ehk)k and a set E with locally finite perimer in Ωsuch that

Ehloc→E in Ω .

If Ω is a bounded open set with C1 boundary, (Eh)h is a sequence of sets with finiteperimeter in Ω and (5.49) holds with Ω′ ≡ Ω, then E is a set with finite perimeter inΩ and

Ehk → E .

Proof. The proof easily follows from Theorem 5.18 and the following exercise.

Exercise: If uh = χEh , u ∈ L1loc(Ω) and

χEh → u in L1loc(Ω) ,

then u = χE Ln-a.e. in Ω, for some measurable set E ⊂ Ω.

Minimal boundaries

De Giorgi [DGCP, Theorem 1.1, Chap. II] (see also [G] and [Mag, Proposition12.29]) proved the existence of minimizers for a Plateau type-problem concerning ageometric variational problem dealing with sets of finite perimeter.

Theorem 5.22 (Existence of minimal boundaries a la De Giorgi, 1960). Let A ⊂ Rn

be a bounded set and let L ⊂ Rn be a measurable set with |∂L|(Rn) < ∞. Then thereexists a solution of the minimization problem

(5.50) min |∂F |(Rn) : F ⊂ Rn measurable, F \ A = L \ ARemark 5.23. In some sense the set L determines the boundary value of F . Roughlyspeaking, prescribing that F \A = L\A we impose L∩∂A as a ”boundary condition”for the admissible sets F . At the same time, the set A, being the region where L canbe modified to minimize perimeter, may act as an obstacle. In general, we do notexpect uniqueness of minimizers for this problem (see [Mag, Sect. 12.5]).

149

Proof of Theorem 5.22. 1st step: Let us denote by

X := F ⊂ Rn : F has locally finite perimeter in Rn, F \ A = L \ ASince L ∈ X, minimization problem (5.50) is equivalent to the problem

(5.51) min |∂F |(Rn) : F ∈ X2nd step: We are going to apply the direct methods of the calculus of variations

for showing the existence of problem (5.51). More precisely we will apply generalizedWeierstrass theorem 5.1 to functional

P : X → [0,∞], P (F ) := |∂F |(Rn)

and we endow the class of competitors X by topology τ = loc of local convergence ofsets in Rn introduced in Definition 5.19 (i). Thus we have to prove that

(5.52) P is sequentially lower semicontinuous ;

(5.53) P is sequentially coercive.

Property (5.52) immediately follows by the lower semicontinuity for sets of finiteperimeter (see Corollary 5.20 (i)). Let us prove (5.53). Fix t ∈ (0,∞) and let

(5.54) (Eh)h ⊂ F ∈ X : P (F ) 6 t .We have to prove there exist a subsequence (Ehk)k and a set E ∈ X such that

(5.55) Ehkloc→E as k →∞ .

Observe thatEh = (Eh ∩ A) ∪ (Eh \ A) ⊆ A ∪ L ∀h .

This implies that, for each open set Ω′ b Rn

(5.56) |Eh ∩ Ω′| 6 |(A ∪ L) ∩ Ω′| ∀h .By (5.54) and (5.56), it follows that (5.49) is fulfilled with Ω = Rn. Thus, by thecompactness of sets of finite perimeter (see Corollary 3.11), (5.55) follows.

IN PROGRESS!

150

References

[A] L. Ambrosio, Corso introduttivo alla Teoria Geometrica della Misura ed alle Superfici min-ime,Appunti dei corsi tenuti da docenti della Scuola, Scuola Normale Siperiore, Pisa,1997.

[A2] L. Ambrosio, La teoria dei perimetri di Caccioppoli-De Giorgi e i suoi piu recenti sviluppi.Rend. Lincei Mat. Appl. 21 (2010), 275–286.

[AFP] L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free DiscontinuityProblems, Oxford Mathematical Monographs, Oxford University Press, 2000.

[AG] L. Ambrosio, R. Ghezzi, Sobolev and bounded variation functions on metric measure spaces,Geometry, Analysis and Dynamics on sub-Riemannian manifolds, vol. II, D. Barilari, U.Boscain, M. Sigalotti edts., EMS Series of Lecture Notes in Mathematics, 2016.

[AK] L. Ambrosio, B. Kirchheim, Rectifiable sets in metric and Banach spaces. Math. Ann. 318,(2000), 527-555.

[AT] L. Ambrosio, P. Tilli, Topics on analysis in metric spaces. Oxford University Press,Oxford,2004.

[Ba] S. Banach, Sur un theoreme de M. Vitali, Fund. Math. 5 (1924), 130–136.[Be1] A.S. Besicovitch, A general form of the covering principle and relative differentiation of addi-

tive functions, Proc.Cambridge Philos.Soc. 41 (1945), 103–110.[Be2] A.S. Besicovitch, A general form of the covering principle and relative differentiation of addi-

tive functions II, Proc. Cambridge Philos. Soc.41 (1946), 1–10.[Be3] A.S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets

of points, Math. Ann. 98 (1928), 422–464.[Be4] A.S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets

of points II, Math. Ann. 115 (1938), 296–329.[Be5] A.S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets

of points III, Math. Ann. 116 (1939), 349–357.[B] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer,

2011. An Italian version is also available: H. Brezis, Analisi Funzionale- Teoria e applicazioni,Liguori, Napoli, 1986.

[C] C. Caratheodory , Vorlesungen uber reelle Funktionen, Leipzig, 1927.

[C2] C. Caratheodory,Uber das lineare Mass von Punktmengen, eine Verallgemeinerung des Langen-begriffs, Nach. Ges. Wiss. Gottingen (1914), 406–426.

[Ch] J. Cheeger Differentiability of Lipschitz functions on metric measure spaces, Geom. Func. Anal.9 (1999),428–517.

[Co] D. L. Cohn, Measure Theory, Birkhauser, 1980.[DG1] E. De Giorgi, Su una teoria generale della misura (r − 1)-dimensionale in uno spazio ad r

dimensioni. Ann.Mat.Pura Appl.(4), 36, (1954), 191–213.[DG2] E. De Giorgi, Nuovi teoremi relativi alle misure (r − 1)-dimensionali in uno spazio ad r

dimensioni,Ricerche Mat. 4, (1955), 95–113.[DG3] E. De Giorgi, Sulla proprieta isoperimetrica dell’ipersfera nella classe degli insiemi aventi

frontiera orientata di misura finita, Memorie Acc. Naz. Lincei Ser. VIII 5, (1958), 33–44.[DGCP] E. De Giorgi, F. Colombini, L.C. Piccinini, Frontiere orientate di misura minima e questioni

collegate. Scuola Normale Superiore, Pisa, 1972.[dG] M. de Guzman, Differentiation of Integrals in Rn, Lecture Notes in Math., Springer-Vrerlag,

1975.[EG] L.C. Evans, R. Gariepy, Lecture Notes on Measure Theory and Fine Properties of Functions,

CRC Press, Boca Raton, 1992.[Fa] K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Second Edi-

tion,Chichester, John Wiley and Sons, 2003.[Fe] H. Federer, Geometric Measure Theory, Springer, 1969.[F] G.B. Folland , Real Analysis. Modern techniques and their applications, Second Edition, Pure

and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons,Inc., New York, 1999.

[GZ] R. F. Gariepy, W. P. Ziemer, Modern Real Analysis, PWS Publishing Company, Boston, 1994.

151

[GO] B. R. Gelbaum, J. M. H. Olmsted Counterexamples in Analysis, Dover Publications,Inc., NewYork, 2003.

[GH] M. Giaquinta, S. Hildebrandt, Calculus of Variations I,Springer, Berlin Heidelberg, 2004.[G] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser, Boston, 1984.[G2] E. Giusti, Direct Methods in the Calculus of Variations, World Scientific, Singapore, 2003.[HOH] H. Hanche-Olsen, H. Holge, The Kolmogorov-Riesz compactness theorem, Expo. Math. 28

(2010), no. 4, 385–394.[Ha] F. Hausdorff, Dimension und ausseres Mass, Math. Ann. 79 (1919), 157–179.[He] J. Heinonen Lectures on analysis on metric spaces, Springer, New York, 2001.[He2] J. Heinonen Lectures on Lipschitz analysis, Report. University of Jyvaskyla Department of

Mathematics and Statistics, 100. University of Jyvaskyla, Jyvaskyla, 2005. ii+77 pp.[LeR] E. Le Donne, S. Rigot, Besicovitch covering property on graded groups and applications to

measure differentiation, Preprint 2015.[Le] H. Lebesgue, Lecons sur l’integration et la recherche de fonctions primitives, Deuxieme Edition,

Gauthier-Villars, Paris, 1926.[Le2] H. Lebesgue, Sur l’integration des fonctions discontinues, Ann. Sci. Ecole Norm. Sup. (3) 27

(1910), 361–450.[Mag] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems, An Introduction to

Geometric Measure Theory , Cambridge University Press, 2012.[MM] U. Massari, M. Miranda Minimal Surfaces of Codimension One, North-Holland, Amsterdam,

1984.[Ma] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press,

1995.[McS] E.J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837–842.[Mi] M. Miranda, Un Teorema di esistenza e unicita per il problema dell’area minima in n variabili.

Ann.Sc.Norm.Sup. Pisa III, 19 (1965), 233–249.[Mo] F. Morgan, Geometric Measure Theory, A beginner’s Guide, Academic Press, 1988.[Mor] A.P. Morse, Perfect blankets, Trans. Amer. Math. Soc. 6 (1947), 418–442.[Ni] O. Nikodym, Sur une genesalization des integrales de M. J. Radon, Fund. Math. 15 (1930),

131–179.[Pe] J. Perez, A New Golden Age of Minimal Surfaces, Notices of the AMS 64 (2017), 347–358.[P] A. Pietsch, History of Banach Spaces and Linear Operators, Birkhauser, Boston, 2007.[Pre] D. Preiss, Dimension of metrics and differentiation of measures, General topology and its

relations to modern analysis and algebra, V (Prague, 1981), Sigma Ser. Pure Math., vol. 3,Heldermann, Berlin, 1983, 565–568.

[Pre2] D. Preiss, Geometry of measures in Rn: distributions. rectifiability, and densities, Ann. ofMath. 125 (1987), 537–643.

[Rad] H. Rademacher,Uber partielle und totale Differenziarbarkeit I, Math. Ann. 79 (1919), 340–359.[Ra] J. Radon, Theorie und Anwendungen der Theorie der absolut additiven Mengenfunktionen,

Sitzungsber. Kaiserl. (Osterreich) Akad. Wiss. Math.- Nat. Kl. , Abtteilung IIa 122, (1913),1295–1438.

[Ro] H. L. Royden, Real Analysis, Macmillan Publishing Company, New York, 1988.[R1] W. Rudin, Real and Complex Analysis, Third Edition, McGraw-Hill Book Co.,New York,1987.[R2] W. Rudin, Functional Analysis, McGraw-Hill Book Co.,New York,1973.[Sch] A.R. Schep, Addendum to ”A still one more proof of the Radon-Nikodym Theorem”, 2006,

available at www.math.sc.edu/ schep/Radon-update.pdf[Ser] R. Serapioni Notes for the course in Geometric Measure Theory-Master degree in Mathematics-

University of Trento.[SC] F. Serra Cassano Notes of Advanced Analysis-a.y. 2016/17-Master degree in Mathematics-

University of Trento.[SC2] F. Serra Cassano, Some topics of geometric measure theory in Carnot groups, Geometry, Anal-

ysis and Dynamics on sub-Riemannian manifolds, vol. I, D. Barilari, U. Boscain, M. Sigalottiedts., EMS Series of Lecture Notes in Mathematics, 2016.

152

[So] S.L. Sobolev, On a theorem of functional analysis (English) , Amer. Math. Soc. Transl. (2) 34(1963), 39–68.

[Ti] J. Tiser, Vitali covering theorem in Hilbert space, Trans. Amer. Math. Soc. 355 (2003), 3277–3289.

[T] F. Treves, Topological vector spaces, distributions and kernels, Academic Press,New York,1967.[Vitali] G. Vitali, Sulle funzioni integrabili, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. 40 (1904-05),

1021–1034.[Vitali2] G. Vitali, Sui gruppi di punti e sulle funzioni di variabili reali, Atti Accad. Sci. Torino 43

(1908), 75–92.

153

INFORMATION ABOUT SOME QUOTED MATHEMATICIANS

Biographical and scientific information more detailed may find at the websitehttp:///www-history.mcs.st-andrews.ac.uk/

• ALAOGLU Leonidas (1914, Red Deer, Alberta, Canada-1981,): Alaoglu was aCanadian-American mathematician, most famous for his widely-cited result calledAlaoglu’s theorem on the weak-star compactness of the closed unit ball in the dualof a normed space, also known as the Banach-Bourbaki-Alaoglu theorem.• BAIRE Rene-Louis (1874, Paris, France- 1932, Chambery, France): Baire worked

on the theory of functions and the concept of a limit. He is best known for the Bairecategory theorem, a result he proved in his 1899 thesis.• BANACH Stefan (1892, Krakow, Austria-Hungary (now Poland) - 1945 in Lvov,

(now Ukraine)): Banach founded modern functional analysis and made major con-tributions to the theory of topological vector spaces. In addition, he contributed tomeasure theory, integration, and orthogonal series.• BESICOVITCH Abram S. (1891, Berdyansk, Russia - 1970, Cambridge, Eng-

land): He was a Russian mathematician, who worked mainly in England. He workedmainly on combinatorial methods and questions in real analysis, and gave importantcontributions in topics such as the Kakeya needle problem, the Hausdorff-Besicovitchdimension and the rectifiability in the plane.• BOREL Emil F.E.J. (1871, Saint Affrique, Aveyron, Midi-Pyrenees, France -

1956, Paris): Borel created the first effective theory of the measure of sets of points,beginning of the modern theory of functions of a real variable.• BOURBAKI Nicolas: Nicolas Bourbaki is the pseudonym of a group of (mainly)

French mathematicians who published an authoritative account of contemporarymathematics.• CACCIOPPOLI Renato (1904, Napoli- 1959, Napoli) was an Italian mathemati-

cian, known for his deep contributions to mathematical analysis, including the theoryof functions of several complex variables, functional analysis, measure theory andpartial differential equations. In particular he pionereed some issues of geometricmeasure theory by means of the introduction of some sets today called Caccioppoli’ssets.• CANTOR George F.L.P. (1845, St Petersburg, Russia - 1918, Halle, Germany):

Cantor founded the set theory and introduced the concept of infinite numbers withhis discovery of cardinal numbers. He also advanced the study of trigonometric series.• CARATHEODORY Constantin (1873, Berlin - 1950, Munich): Caratheodory

made significant contributions to the calculus of variations, the theory of point setmeasure, and the theory of functions of a real variable.• CAUCHY Augustin-Louis (1789, Paris, France - 1857, Sceaux (near Paris),

France) Cauchy pioneered the study of analysis, both real and complex, and thetheory of permutation groups. He also researched in convergence and divergenceof infinite series, differential equations, determinants, probability and mathematicalphysics.• DE GIORGI Ennio (1928, Lecce, Italy - 1996, Pisa, Italy): he was an Italian

mathematician, who worked on calculus of variations, geometric measure theory, par-tial differential equations and the foundations of mathematics, giving fundamental

154

contributions. In particular he did the last step for solving the 19th Hilbert problemon the regularity of solutions of elliptic partial differential equations (togheter with J.F. Nash, but independently) and solved the so-called Bernstein problem for minimalsurfaces (in collaboration with E. Bombieri and E. Giusti).

• DE LA VALLEE POUSSIN Charles J. (1866, Louvain, Belgio-1962, Louvain,Belgio): he was a Belgian mathematician and is best known for proving the primenumber theorem.• DIRICHLET Gustav L. (1805, Duren, French Empire (now Germany)- 1859,

Gottingen, Hanover (now Germany)) He made valuable contributions to number the-ory, analysis, and mechanics. In number theory he proved the existence of an infinitenumber of primes in any arithmetic series. In mechanics he investigated the equi-librium of systems and potential theory, which led him to the Dirichlet problemconcerning harmonic functions with prescribed boundary values.• EGOROFF Dimitri F. (1869, Moscow, Russia - 1931, Kazan, USSR): He worked

on integral equations and a theorem in the theory of functions of real variable isnamed after him. Luzin was Egorov’s first student and became a member of theschool Egorov created in Moscow dealing with functions of real variable.• EULER Leonhard (1707, Basel, Switzerland -1783, St Petersburg, Russia) He was

a pioneering Swiss mathematician and physicist. He made important discoveries infields as diverse as infinitesimal calculus and graph theory. He also introduced muchof the modern mathematical terminology and notation, particularly for mathematicalanalysis, such as the notion of a mathematical function.He is also renowed for his workin mechanics, fluid dynamics, optics, and astronomy.• FATOU Pierre J.L. (1878, Lorient, France - 1929, Pornichet, France): Fatou

worked in the fields of complex analytic dynamic and iterative and recursive processes.• FISHER Ernst (1875, Vienna, Austria - 1954, Cologne, Germany): Ernst Fischer

is best known for the Riesz-Fischer theorem in the theory of Lebesgue integration.• FOURIER Joseph J.B. (1768, Auxerre, Francia - 1830, Parigi): Fourier studied

the mathematical theory of heat conduction. He established the partial differentialequation governing heat diffusion and solved it by using infinite series of trigonometricfunctions.• FRECHET Maurice (1878, Maligny - 1973, Paris) Frechet was a French math-

ematician who made major contributions to the topology of point sets and definedand founded the theory of abstract spaces.In particular, in his thesis he introducedthe concept of a metric space, although he did not invent the name ’metric space’which is due to Hausdorff.• FRIEDRICHS Kurt Otto (1901, Kiel, Germany- 1982, New Rochelle, New York,

USA) Friedrichs’ greatest contribution to applied mathematics was his work on partialdifferential equations. He also did major research and wrote many books.• FUBINI Guido, (1879, Venice, Italy -1943, New York, USA) Fubini may be

considered one of the founder of modern projective-differential geometry. He hasbeen very important as an analyst (Lebesgue integrals) and in the study of automor-phic functions and discontinuous groups. He has been also engaged in mathematicalphysics and in applied mathematics.• GAUSS Carl F. (1777, Brunswick, Duchy of Brunswick (now Germany) - 1855,

Gottingen, Hanover (now Germany)) Gauss worked in a wide variety of fields in both

155

mathematics and physics incuding number theory, analysis, differential geometry,geodesy, magnetism, astronomy and optics. His work has had an immense influencein many areas.•GREEN George (1793, Sneinton, Nottingham, England - 1841, Sneinton, Notting-

ham, England) George Green was an English mathematician best-known for Green’sfunction and Greeen’s theorems in potential theory.• HADAMARD Jacques S. (1865, Versailles, Francia - 1963, Parigi)???• HAHN Hans (1879, Vienna, Austria - 1934, Vienna, Austria): Hahn was an

Austrian mathematician who is best remembered for the Hahn-Banach theorem. Healso made important contributions to the calculus of variations, developing ideas ofWeierstrass.• HAUSDORFF Felix (1868, Breslau, Germany (now Wroclaw, Poland)- 1942,

Bonn, Germany ): Hausdorff worked in topology creating a theory of topological andmetric spaces. In particular, he introduced the modern notion of metric space. Healso worked in set theory and introduced the concept of a partially ordered set.• HILBERT David (1862, Konigsberg, Prussia (now Kaliningrad, Russia)-1943,

Gottingen, Germany): Hilbert’s work in geometry had the greatest influence in thatarea after Euclid. A systematic study of the axioms of Euclidean geometry led Hilbertto propose 21 such axioms and he analysed their significance. He made contributionsin many areas of mathematics and physics.• HOLDER Otto L. (1859, Stuttgart, Germany - 1937, Leipzig, Germany): Holder

worked on the convergence of Fourier series and in 1884 he discovered the inequalitynow named after him. He became interested in group theory through Kronecker andKlein and proved the uniqueness of the factor groups in a composition series.• KOLMOGOROV Andrey N. (1903, Tambov, Tambov province, Russia - 1987,

Moscow) He was a Soviet Russian mathematician, preeminent in the 20th century,who advanced various scientific fields, among them probability theory, topology, in-tuitionistic logic, turbulence, classical mechanics and computational complexity.• JORDAN Camille M.E. (1838, La Croix-Rousse, Lyon, France - 1922, Paris,

France): Jordan was highly regarded by his contemporaries for his work in algebra,group theory and Galois theory. Jordan is best remembered today among analystsand topologists for his proof that a simply closed curve divides a plane into exactlytwo regions, now called the Jordan curve theorem. He also originated the concept offunctions of bounded variation and is known especially for his definition of the lengthof a curve.• LAGRANGE Joseph-Louis (1736, Turin, Sardinia-Piedmont (now Italy) - 1813 in

Paris, France) Born Giuseppe Lodovico (Luigi) Lagrangia, he was a mathematicianand astronomer, lived part of his life in Prussia and part in France, making greatcontributions to all fields of analysis, to number theory, and to classical and celestialmechanics. In particular he was one of the founders of the caculus of variations.• LEBESGUE Henry L. (1875, Beauvais, Oise, Picardie, France-1941, Paris, France):

Lebesgue formulated the theory of measure in 1901 and the following year he gavethe definition of the Lebesgue integral that generalises the notion of the Riemannintegral.

156

• LEVI Beppo (1875, Turin, Italy - 1961, Rosarno, Argentina): He studied singu-larities on algebraic curves and surfaces. Later he proved some foundational resultsconcerning Lebesgue integration.• LIPSCHITZ Rudolf O.S.( 1832, Konisberg, Germany (now Kaliningrad, Russia)

-1903, Bonn, Germany) He was a German mathematician and professor at the Uni-versity of Bonn from 1864. Dirichlet was his teacher. While Lipschitz gave his nameto the Lipschitz continuity condition, he worked in a broad range of areas. Theseincluded number theory, algebras with involution, mathematical analysis, differentialgeometry and classical mechanics.• LUSIN Nikolai N. (1883, Irkutsk, Russia - 1950, Moscow, USSR): Lusin’s main

contributions are in the area of foundations of mathematics and measure theory. Healso made significant contributions to descriptive set topology.• MINKOWSKI Hermann (1864, Alexotas, Russian Empire (now Kaunas, Lithua-

nia) - 1909, Gottingen, Germany): Minkowski developed a new view of space andtime and laid the mathematical foundation of the theory of relativity.• MORSE Anthony P. (1911-1984): he was an American mathematician who

worked in both analysis, especially measure theory, and in the foundations of math-ematics. He is best known as the co-creator, together with John L. Kelley, of Morse-Kelley set theory. This theory first appeared in print in Kelley’s General Topology.He is also known for his work on the Morse-Sard theorem and the Federer-Morsetheorem.• NIKODYM Otto M. (1887, Zablotow, Galicia, Austria-Hungary (now Ukraine)

- 1974, Utica, USA ): Nikodym’s name is mostly known in measure theory (e. g.the Radon-Nikodym theorem and derivative, the Nikodym convergence theorem, theNikodym-Grothendieck boundedness theorem), in functional analysis (the Radon-Nikodym property of a Banach space, the Frechet-Nikodym metric space, a Nikodymset), projections onto convex sets with applications to Dirichlet problem, general-ized solutions of differential equations, descriptive set theory and the foundations ofquantum mechanics.• PEANO Giuseppe (1858, Cuneo, Italy - 1932, Turin, Italy): Peano was the

founder of symbolic logic and his interests centred on the foundations of mathematicsand on the development of a formal logical language. Among his important contri-butions, let us recall he invented ’space-filling’ curves in 1890, these are continuoussurjective mappings from [0,1] onto the unit square.• PLATEAU Joseph A. F. (Tournal, Belgium, 1801- Ghent, Belgium, 1883): he

was a Belgian physicist, who studied the phenomena of capillary action and surfacetension. The mathematical problem of existence of a minimal surface with a givenboundary is named after him Plateau’s problem. He conducted extensive studies ofsoap films and formulated Plateau’s laws which describe the structures formed bysuch films in foams.• POINCARE Jules Henri (1854, Nancy, Meurthe-et-Moselle - 1912, Paris) He was

a French mathematician, theoretical physicist, and a philosopher of science. He isoften described as a polymath, and in mathematics as The Last Universalist, sincehe excelled in all fields of the discipline as it existed during his lifetime. As a mathe-matician and physicist, he made many original fundamental contributions to pure and

157

applied mathematics, mathematical physics, and celestial mechanics. He is consideredto be one of the founders of the field of topology.• POISSON S. Denis (1781, Pithiviers, France - 1840, Sceaux near Paris) He was

very well-known for his work on definite integrals, electromagnetic theory, and prob-ability. PoissonOs most important work concerned the application of mathematicsto electricity and magnetism, mechanics, and other areas of physics. Poisson con-tributed to celestial mechanics by extending the work of Lagrange and Laplace onthe stability of planetary orbits and by calculating the gravitational attraction ex-erted by spheroidal and ellipsoidal bodies. He also did important investigation ofprobability. In pure mathematics his most important works were a series of paperson definite integrals and his advances in Fourier analysis, which paved the way forthe research of the German mathematicians Peter Dirichlet and Bernhard Riemann.• PRYM Friedrich E.F. (1841 Duren - 1915 Bonn) He was a German mathematician

who introduced Prym varieties and Prym differentials.• RADEMACHER Hans (1892, Wandsbeck, now Hamburg-Wandsbek - 1969, Haver-

ford, Pennsylvania, USA) Rademacher performed research in analytic number theory,mathematical genetics, the theory of functions of a real variable, and quantum theory.Most notably, he developed the theory of Dedekind sums. Rademacher’s name is alsoknown for the result about the differentiability of Lipschitz functions.• RADON Johann (1887, Tetschen, Bohemia (now Decin, Czech Republic) - 1956,

Vienna, Austria): Radon worked on the calculus of variations, differential geometryand measure theory.• RIEMANN G. F. Bernhard (1826, Breselenz, Hanover (now Germany)- 1866,

Selasca, Italy): Riemann’s ideas concerning geometry of space had a profound effecton the development of modern theoretical physics. He clarified the notion of integralby defining what we now call the Riemann integral.• RIESZ Frigyes (Friedrich) (1880, Gyor, Austria-Hungary (now Hungary) - 1956,

Budapest, Hungary ): Riesz was a founder of functional analysis and his work hasmany important applications in physics.• RIESZ Marcel (1886, Gyor, Austria-Hungary (now Hungary) - 1969, Lund, Swee-

den) He was a Hungarian mathematician and moved to Sweden in 1908 and spentthe rest of his life there. He was known for work on classical analysis, on fundamentalsolutions of partial differential equations, on divergent series, Clifford algebras, andnumber theory. He was the younger brother of the mathematician Frigyes Riesz.• SCHAUDER Juliusz P. (1899, Lemberg, Austrian Empire (now Lviv, Ukraine) -

1943, Lwow, Poland (now Ukraine)) He was a Polish mathematician of Jewish origin,known for his fundamental work in functional analysis, partial differential equationand mathematical physics. Schauder was Jewish, and after the invasion of Germantroops in Lwow it was impossible for him to continue his work. He was executed bythe Gestapo, probably in September 1943.• SERRIN James (1926, Chicago, Illinois, USA - 2012, Minneapolis, Minnesota,

USA) Serrin is a mathematician well-known for his contributions to continuum me-chanics, nonlinear analysis, and partial differential equations.• SOBOLEV Sergei L. (1908, S.Petersburg - 1989, Moscow): He introduced the

notions that are now fundamental for several different areas of mathematics. Sobolevspaces and their embedding theorems are an important subject in functional analysis.

158

• STEINHAUS Hugo D. (1887, Jaslo, Galicia, Austrian Empire (now Poland) -1972, Wroclaw, Poland): He did important work on functional analysis. Some ofSteinhaus’s early work was on trigonometric series. He was the first to give someexamples which would lead to marked progress in the subject.• TONELLI Leonida (1885, Gallipoli, Italy - 1946, Pisa, Italy) Tonelli discovered

the importance of the semicontinuity in calculus of variations in order to get theexistence of minima or maxima for functionals. He also advanced the study of theintegration theory.•URYSOHN Pavel S. (1898, Odessa, Ukraine- 1924, Batz-sur-Mer, France): Urysohn

is best known for his contributions in the theory of dimension, and for Urysohn’sMetrization Theorem and Urysohn’s Lemma, both of which are fundamental resultsin topology.• VITALI Giuseppe (1875, Ravenna, Italy - 1932, Bologna, Italy): Vitali made

significant mathematical discoveries including a theorem on set-covering, the notionand the characterization of an absolutely continuous functions and a criterion for theclosure of a system of orthogonal functions.• VOLTERRA Vito (1860, Ancona, Italy - 1940, Roma, Italy) Volterra was an Ital-

ian mathematician and physicist, known for his contributions to mathematical biologyand integral equations, being one of the founders of functional analysis. Volterra isthe one among few people who was a plenary speaker in the International Congressof Mathematicians four times (1900, 1908, 1920, 1928). In 1922, he joined the oppo-sition to the Fascist regime of Benito Mussolini and in 1931 he was one of only 12 outof 1,250 professors who refused to take a mandatory oath of loyalty. As a result ofhis refusal to sign the oath of allegiance to the fascist government, he was compelledto resign his university post and his membership of scientific academies, and, duringthe following years, he lived largely abroad, returning to Rome just before his death.• VON NEUMANN John (1903, Budapest, Hungary - 1957, Washington D.C.,

USA): Von Neumann was generally regarded as the foremost mathematician of histime and said to be ”the last representative of the great mathematicians”;a geniuswho was comfortable integrating both pure and applied sciences. He made majorcontributions to a number of fields, including mathematics (foundations of mathe-matics, functional analysis, ergodic theory, representation theory, operator algebras,geometry, topology, and numerical analysis), physics (quantum mechanics, hydrody-namics, and quantum statistical mechanics), economics (game theory), computing(Von Neumann architecture, linear programming, self-replicating machines, stochas-tic computing), and statistics.He was a pioneer of the application of operator theoryto quantum mechanics in the development of functional analysis, and a key figure inthe development of game theory and the concepts of cellular automata, the universalconstructor and the digital computer.• WEIERSTRASS Karl (1815, Ostenfelde, Germania -1897, Berlino): Weierstrass

is best known for his construction of the theory of complex functions by means ofpower series. Known as the father of modern analysis, Weierstrass devised tests forthe convergence of series and contributed to the theory of periodic functions, functionsof real variables, elliptic functions, Abelian functions, converging infinite products,and the calculus of variations.

Documents

AN INTRODUCTION TO GEOMETRIC MEASURE THEORY AND AN …€¦ · AN INTRODUCTION TO GEOMETRIC MEASURE THEORY AND AN APPLICATION TO MINIMAL SURFACES ( DRAFT DOCUMENT) Academic Year 2018/19