From length, area, volume to measure l Ý N Ý length... · From length, area, volume to measure l Ý!¡¨!N¨ ÿÝ The concepts of length, area, volume Stage 1: classical Euclidean

From length, area, volume to measure l�Ý!¡È!NÈ�ÿÝ

From length, area, volume to measurel�Ý!¡È!NÈ�ÿÝ

Mijia Lai £5�\¤

[email protected]


Outline

1. History

2. Lebesgue measure

3. Hausdorff measure

4. Isodiametric inequality

5. Projects


The concepts of length, area, volume

Stage 1: classical Euclidean geometry (earlier than 400 B.C.)

Ruler: measure length of straight lines.

Gilded Bronze Ruler - 1 chi = 23.1 cm. Western Han 206 B.C.

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Stage 1: classical Euclidean geometry (earlier than 400 B.C.)Ruler: measure length of straight lines.



=⇒4��p��180cm.






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Stage 1

We know the area for triangles, rectangles, then for polygons.Volumes of cube, polytope, ball, cylinder, etc. All shapes involvedare nice and regular.

Left: Fields Medal; Right: Archimedes’ cylinder and sphere


Stage 1

We know the area for triangles, rectangles, then for polygons.Volumes of cube, polytope, ball, cylinder, etc. All shapes involvedare nice and regular.

Left: Fields Medal; Right: Archimedes’ cylinder and sphere


Chinese contributions

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Chinese contributions

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Stage 2

Stage 2: Entering Calculus (from 17th century)Something you learned in last two semesters.

What?

Arc length, volume, surface area of revolution, etc. Shapes are stillnice and regular.


Stage 2


What?



Stage 2


What?



Stage 3

Stage 3: Introduction of measure due to Peano, Jordan, Borel,Lebesgue, Caratheodory and many others. (late 19th century)

Goal: To generalize the notions of length, area, volume to treatmore general subsets of Rn.


Ideas

For a nonnegative Riemannian integrable function f ,∫ b

af (x)dx

represents the area under the graph of f (x).

Question: why this is true and accurate?


Ideas

For a nonnegative Riemannian integrable function f ,∫ b

af (x)dx

represents the area under the graph of f (x).

Question: why this is true and accurate?


The answer relies on a priori the definition of area. In fact, thefollowing ’squeeze theorem’ justifies that we must define the areain this way and it is accurate quantitatively.

∑mi∆xi ≤

∫ b

af (x)dx ≤

∑Mi∆xi ,

where mi (Mi ) = infx∈[xi−1,xi ](supx∈[xi−1,xi ])f (x).

Based on the intuition that area should be monotonic with respectto the set inclusion, ’Squeeze theorem’ gives the accurate accountof the area under the curve.


The answer relies on a priori the definition of area. In fact, thefollowing ’squeeze theorem’ justifies that we must define the areain this way and it is accurate quantitatively.

∑mi∆xi ≤

∫ b

af (x)dx ≤

∑Mi∆xi ,

where mi (Mi ) = infx∈[xi−1,xi ](supx∈[xi−1,xi ])f (x).

Based on the intuition that area should be monotonic with respectto the set inclusion, ’Squeeze theorem’ gives the accurate accountof the area under the curve.


Sometimes, it is difficult to have this kind of two sided ’squeeze’.Nevertheless the completeness of real number shows that one sidedsqueeze produces meaningful quantities as well.

Arc length: A parametric curve Γ(t) = (x(t), y(t), z(t)), t ∈ [0, 1].Arc length is first defined as

l(Γ) = sup{k∑

i=1

|Γ(ti )−Γ(ti−1)||0 = t0 < t1 < · · · < tk = 1,∀k ∈ N}.

Of course, it is only meaningful to study curves which are of finitelength (at least locally), they are called rectifiable curves.


Sometimes, it is difficult to have this kind of two sided ’squeeze’.Nevertheless the completeness of real number shows that one sidedsqueeze produces meaningful quantities as well.

Arc length: A parametric curve Γ(t) = (x(t), y(t), z(t)), t ∈ [0, 1].Arc length is first defined as

l(Γ) = sup{k∑

i=1

|Γ(ti )−Γ(ti−1)||0 = t0 < t1 < · · · < tk = 1,∀k ∈ N}.

Of course, it is only meaningful to study curves which are of finitelength (at least locally), they are called rectifiable curves.


Peano curve

A continuous parametric curve Γ, whose image is the whole square.http://demonstrations.wolfram.com/PeanoCurveExplorer/

http://demonstrations.wolfram.com/PeanoCurveExplorer/


Generally, given a subset A ⊂ Rn, one first needs to give up thetwo sided approximation. Secondly, any meaningful definitionshould satisfy our intuition and coincide with classical notionswhen restricting to classical objects:

1. Generalization of classical notion of length (n = 1), area(n = 2), volume (n ≥ 3)

2. Monotonicity

3. Translation invariant

4. Additivity (countably) /��q�§±�uØ��0

A set consists of infinity elements is called countable if there is abijection from the set to N.Examples: Q is countable, [0, 1] is not.


Generally, given a subset A ⊂ Rn, one first needs to give up thetwo sided approximation. Secondly, any meaningful definitionshould satisfy our intuition and coincide with classical notionswhen restricting to classical objects:

1. Generalization of classical notion of length (n = 1), area(n = 2), volume (n ≥ 3)

2. Monotonicity

3. Translation invariant

4. Additivity (countably) /��q�§±�uØ��0

A set consists of infinity elements is called countable if there is abijection from the set to N.Examples: Q is countable, [0, 1] is not.


Outer measure

Given A ⊂ Rn, let

m∗(A) := inf{∞∑i=1

|Ti |,A ⊂∞⋃i=1

Ti},

where the inf is taken over all possible coverings of A by countablymany rectangles Ti in Rn. |Ti | is the volume of rectangle, which isof course the product of length of all mutually orthogonal sides.

This is approximation from outer or above. This time, we have ingeneral no approximation from inner or below. That is what wehave to give up. But by taking inf of all possible upper bounds,intuitively the definition makes good sense.


Outer measure

Given A ⊂ Rn, let

m∗(A) := inf{∞∑i=1

|Ti |,A ⊂∞⋃i=1

Ti},

where the inf is taken over all possible coverings of A by countablymany rectangles Ti in Rn. |Ti | is the volume of rectangle, which isof course the product of length of all mutually orthogonal sides.

This is approximation from outer or above. This time, we have ingeneral no approximation from inner or below. That is what wehave to give up. But by taking inf of all possible upper bounds,intuitively the definition makes good sense.


However, there do exist sets which violate the countable additivityunder this framework. The remedy is to work with only those setswhich satisfy countable additivity and call them measurable sets,and outer measure restricted to these sets is simply called themeasure (Lebesgue measure).

Round or straight figures → Regular shapes → measurable sets.


However, there do exist sets which violate the countable additivityunder this framework. The remedy is to work with only those setswhich satisfy countable additivity and call them measurable sets,and outer measure restricted to these sets is simply called themeasure (Lebesgue measure).

Round or straight figures → Regular shapes → measurable sets.


More properties and remarks

I Measurable sets are quite many, the usual sets like open sets,closed sets, their countable unions, intersections, are allmeasurable.

I One can replace rectangles in the definition by othermeasurable sets.

I Homogeneity: m(λA) = λnm(A), ∀λ > 0.

I The Lebesgue measure only measures the top dimensionalcontent of a set in Rn.


Question: What could we do to manifest the lower dimensionalcontent?

Recall:

m(A) := inf{∞∑i=1

|Ti |,A ⊂∞⋃i

Ti},

Instead of using Ti and its top volume |Ti |, we could try any setwhich is of ’lower dimensional’ in nature, and use its lowerdimensional substitute for |Ti |.


Question: What could we do to manifest the lower dimensionalcontent?

Recall:

m(A) := inf{∞∑i=1

|Ti |,A ⊂∞⋃i

Ti},

Instead of using Ti and its top volume |Ti |, we could try any setwhich is of ’lower dimensional’ in nature, and use its lowerdimensional substitute for |Ti |.


Some key ingredients

Volume of n-dimensional ball:

Vn(r) =π

n2

Γ(n2 + 1)rn := α(n)rn,

where Γ(s) =∫∞0 e−xx s−1dx . The key is to observe the right hand

side can be defined for any real number n, not necessarily forintegers.

Isodiametric inequality: for any measurable set A ⊂ Rn,

|A| ≤ α(n)(diam(A)

2)n.

Diameter of a set A: diam(A) := supx ,y∈A |x − y |.


Some key ingredients

Volume of n-dimensional ball:

Vn(r) =π

n2

Γ(n2 + 1)rn := α(n)rn,

where Γ(s) =∫∞0 e−xx s−1dx . The key is to observe the right hand

side can be defined for any real number n, not necessarily forintegers.

Isodiametric inequality: for any measurable set A ⊂ Rn,


2)n.

Diameter of a set A: diam(A) := supx ,y∈A |x − y |.


These observations inspire two directions to generalize thedefinition of Lebesgue measure: 1. using arbitrary sets instead ofrectangles; 2. using the quantity α(n)(diam(A)

2 )n as the volume ofA.Fix two parameters δ > 0 and s ≥ 0, given A ⊂ Rn define:

Hsδ (A) = inf{

∞∑i=1

α(s)(diam(Ri )

2)s , diam(Ri ) ≤ δ,A ⊂

∞⋃i=1

Ri}.

Clearly, Hsδ is non-decreasing as δ → 0+, let

Hs(A) := limδ→0

Hsδ (A).

This is called the s-Hausdorff measure of A.


These observations inspire two directions to generalize thedefinition of Lebesgue measure: 1. using arbitrary sets instead ofrectangles; 2. using the quantity α(n)(diam(A)

2 )n as the volume ofA.Fix two parameters δ > 0 and s ≥ 0, given A ⊂ Rn define:

Hsδ (A) = inf{

∞∑i=1

α(s)(diam(Ri )

2)s , diam(Ri ) ≤ δ,A ⊂

∞⋃i=1

Ri}.

Clearly, Hsδ is non-decreasing as δ → 0+, let

Hs(A) := limδ→0

Hsδ (A).

This is called the s-Hausdorff measure of A.


Properties and remarks

I If 0 < Hs(A) ≤ ∞, then Ht(A) =∞, ∀t < s, if0 ≤ Hs(A) <∞, then Ht(A) = 0, ∀t > s.

I Lets = sup{t|Ht(A) =∞} = inf{t|Ht(A) = 0},

this s is called the Hausdorff dimension of A, denoted bydimH(A). s is not necessarily an integer.

I dimH(A) ≤ n, ∀A ⊂ Rn.

I In Rn, Hn is same as the Lebesgue measure.

I Homogeneity: Hs(λA) = λsHs(A).

I Countable additivity.


Examples of sets which are fractional dimension

Cantor set

2Hs(A) = Hs(3A) = 3sHs(A),

implies that s = log3 2.


Examples of sets which are fractional dimension

Cantor set

2Hs(A) = Hs(3A) = 3sHs(A),



Von Koch curve

Von Koch curve

4Hs(A) = Hs(3A) = 3sHs(A),



Von Koch curve

Von Koch curve

4Hs(A) = Hs(3A) = 3sHs(A),



Isodiametric inequality

Theorem (Bieberbach)

For any measurable set A ⊂ Rn,


2)n.


Steiner symmetrization


Claim: diam(Ae) ≤ diam(A)£��½n¤ ; |Ae | = |A| £y3�n¤.




Claim: diam(Ae) ≤ diam(A)£��½n¤ ; |Ae | = |A| £y3�n¤.


Taking an orthornormal basis {e1, · · · , en}, perform n-thsymmetrization, let A∗ be the resulted domain, then

diam(A∗) ≤ diam(A), |A ∗ | = |A|.

However after n-th symmetrization, A∗ is symmetric about origin,thus ∀x ∈ A∗, 2|x | ≤ diam(A∗), which implies

A∗ ⊂ B diam(A∗)2

(0).

Hence

|A| = |A ∗ | ≤ α(n)(diam(A∗)

2)n ≤ α(n)(

diam(A)

2)n.


Selected topics for further readings

1. c�È©��È©g�

2. On Peano’s space-filling curves

3. Hausdorff dimension V.S. topological dimension

4. Minkowski content

5. Steiner symmetrization

Documents

From length, area, volume to measure l Ý N Ý length... · From length, area, volume to measure l Ý!¡¨!N¨ ÿÝ The concepts of length, area, volume Stage 1: classical Euclidean