Lebesgue Integral

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Tutorial 5: Lebesgue Integration 1

5. Lebesgue Integration

In the following, (, F, ) is a measure space.

Definition 39 LetA . We callcharacteristic function ofA,the map 1A: R, defined by:

, 1A()=

1 if A0 if A

Exercise 1. Given A , show that 1A : (, F) (R, B(R)) ismeasurable if and only ifA F.

Definition 40 Let(, F) be a measurable space. We say that a maps : R+ is a simple function on (, F), if and only if s is ofthe form :

s=ni=1

i1Ai

wheren 1, iR+ andAi F, for all i= 1, . . . , n.

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Exercise 2. Show that s : (, F) (R+, B(R+)) is measurable,whenevers is a simple function on (, F).

Exercise 3. Letsbe a simple function on (, F) with representations =

ni=1i1Ai. Consider the map : {0, 1}

n defined by() = (1A1(), . . . , 1An()). For each y s(), pick one y such thaty =s(y). Consider the map:s() {0, 1}n defined by(y) =(y).

1. Show that is injective, and that s() is a finite subset ofR+.

2. Show thats=

s()1{s=}

3. Show that any simple functions can be represented as:

s=

ni=1

i1Ai

wheren 1, iR+, Ai Fand =A1 . . . An.

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Definition 41 Let(, F) be a measurable space, ands be a simplefunction on (, F). We callpartition of the simple functions, any

representation of the form:

s=ni=1

i1Ai

wheren 1, iR+, Ai F and =A1 . . . An.

Exercise 4. Letsbe a simple function on (, F) with two partitions:

s=n

i=1

i1Ai =m

j=1

j1Bj

1. Show thats=

i,ji1AiBj is a partition ofs.

2. Recall the convention 0(+) = 0 and (+) = +if > 0. For all a1, . . . , ap in [0, +], p 1 and x [0, +],

prove the distributive property: x(a1+. . .+ap) =xa1+. . .+xap.

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3. Show thatn

i=1i(Ai) =m

j=1j(Bj).

4. Explain why the following definition is legitimate.

Definition 42 Let (, F, ) be a measure space, ands be a simplefunction on(, F). We define theintegral ofs with respect to , as

the sum, denotedI(s), defined by:

I(s)=

ni=1

i(Ai) [0, +]

wheres=

ni=1i1Ai

is any partition ofs

.

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Exercise 5. Lets, tbe two simple functions on (, F) with partitionss=

ni=1i1Ai and t=

mj=1j1Bj . Let R

+.

1. Show thats+t is a simple function on (, F) with partition:

s+t=ni=1

mj=1

(i+j)1AiBj

2. Show thatI(s+t) =I(s) +I(t).

3. Show thats is a simple function on (, F).

4. Show thatI(s) =I(s).

5. Why is the notationI(s) meaningless if= +or


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Exercise 6. Let f : (, F) [0, +] be a non-negative and mea-

surable map. For all n1, we define:

sn=

n2n1k=0

k

2n1{ k

2nf< k+1

2n }+n1{nf} (1)

1. Show thatsn is a simple function on (, F), for all n 1.

2. Show that equation (1) is a partition sn, for all n 1.

3. Show thatsn sn+1f, for all n1.

4. Show thatsn f as n +1.

1 i.e. for all , the sequence (sn())n1 is non-decreasing and convergesto f() [0, +].

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Theorem 18 Letf : (, F)[0, +] be a non-negative and mea-surable map, where (, F) is a measurable space. There exists a se-

quence(sn)n1 of simple functions on(, F) such thatsnf.

Definition 43 Let f : (, F) [0, +] be a non-negative andmeasurable map, where (, F, ) is a measure space. We define theLebesgue integral off with respect to , denoted

f d, as:

f d = sup{I(s) : s simple function on(, F) , s f}

where, given any simple functions on(, F), I(s) denotes its inte-gral with respect to .

Exercise 7. Let f : (, F) [0, +] be a non-negative and mea-surable map.

1. Show that

f d [0, +].

2. Show that

f d= I(f), wheneverfis a simple function.

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3. Show that

gd

f d, whenever g : (, F) [0, +] isnon-negative and measurable map with gf.

4. Show that

(cf)d = c

f d, if 0 < c < +. Explain whyboth integrals are well defined. Is the equality still true forc= 0.

5. For n 1, put An = {f > 1/n}, and sn = (1/n)1An. Show

that sn is a simple function on (, F) with sn f. Show thatAn {f >0}.

6. Show that

f d= 0 ({f >0}) = 0.

7. Show that ifs is a simple function on (,F

) with s

f, then({f >0}) = 0 implies I(s) = 0.

8. Show that

f d= 0 ({f >0}) = 0.

9. Show that (+)f d = (+) f d. Explain why both inte-grals are well defined.

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10. Show that (+)1{f=+}f and:

(+)1{f=+}d= (+)({f= +})

11. Show that

fd


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3. Let :F [0, +] be defined by (A) =I(s1A). Show thatis a measure on F.

4. SupposeAn F, AnA. Show that I(s1An)I(s1A).

Exercise 9.Let (fn)n1 be a sequence of non-negative and measur-

able maps fn: (, F) [0, +], such that fnf.1. Recall what the notationfnf means.

2. Explain why f : (, F) (R, B(R)) is measurable.

3. Let = supn1

fnd. Show that

fnd .4. Show that

f d.

5. Let s be any simple function on (, F) such that s f. Letc]0, 1[. For n 1, define An ={cs fn}. Show that An F

and An .

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6. Show thatcI(s1An)

fnd, for alln 1.

7. Show thatcI(s) .

8. Show thatI(s) .

9. Show that

f d .

10. Conclude that fnd f d.

Theorem 19 (Monotone Convergence) Let(, F, ) be a mea-sure space. Let(fn)n1 be a sequence of non-negative and measurable

mapsfn: (, F) [0, +] such thatfnf. Then

fnd

f d.

Exercise 10. Let f, g : (, F) [0, +] be two non-negative andmeasurable maps. Let a, b [0, +].

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1. Show that if (fn)n1 and (gn)n1 are two sequences of non-negative and measurable maps such that fn f and gn g,

then fn+gnf+g.2. Show that

(f+g)d=

f d+

gd.

3. Show that

(af+bg)d= a

f d+b

gd.

Exercise 11. Let (fn)n1 be a sequence of non-negative and mea-

surable maps fn: (, F) [0, +]. Define f=+

n=1fn.

1. Explain why f : (, F)[0, +] is well defined, non-negative

and measurable.

2. Show that

f d=+

n=1

fnd.

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Definition 44 Let (, F, ) be a measure space and let P() be aproperty depending on . We say that the property P() holds

-almost surely, and we writeP() -a.s., if and only if:N F , (N) = 0 , Nc, P() holds

Exercise 12.LetP() be a property depending on , such that

{ :P() holds}is an element of the -algebraF.1. Show thatP() , -a.s. ({ :P() holds}c) = 0.

2. Explain why in general, the right-hand side of this equivalencecannot be used to defined -almost sure properties.

Exercise 13. Let (, F, ) be a measure space and (An)n1 be a

sequence of elements ofF. Show that(+n=1An)+

n=1(An).

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Exercise 14.Let (fn)n1 be a sequence of maps fn : [0, +].

1. Translate formally the statement fnf -a.s.

2. Translate formallyfn f -a.s. andn, (fn fn+1 -a.s.)

3. Show that the statements 1.and 2.are equivalent.

Exercise 15. Suppose that f, g : (, F) [0, +] are non-negativeand measurable with f = g -a.s.. LetN F, (N) = 0 such thatf = g on Nc. Explain why

f d =

(f1N)d+

(f1Nc)d, all

integrals being well defined. Show that f d= gd.Exercise 16. Suppose (fn)n1 is a sequence of non-negative andmeasurable maps and fis a non-negative and measurable map, suchthat fn f -a.s.. Let N F, (N) = 0, such that fn f on Nc.Define fn=fn1Nc and f=f1Nc .

1. Explain why fand the fns are non-negative and measurable.

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2. Show that fn f.

3. Show that fnd f d.

Exercise 17.Let (fn)n1be a sequence of non-negative and measur-able mapsfn: (, F) [0, +]. Forn 1, we definegn= infknfk.

1. Explain why thegns are non-negative and measurable.

2. Show thatgnlim inffn.

3. Show that

gnd

fnd, for all n 1.

4. Show that if (un)n1 and (vn)n1 are two sequences in R withunvn for all n 1, then lim infunlim infvn.

5. Show that

(lim inffn)d lim inf

fnd, and recall why allintegrals are well defined.

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Theorem 20 (Fatou Lemma) Let (, F, ) be a measure space,and let (fn)n1 be a sequence of non-negative and measurable maps

fn: (, F) [0, +]. Then: (lim infn+

fn)dlim infn+

fnd

Exercise 18.Let f : (, F)[0, +] be a non-negative and mea-

surable map. Let A F.

1. Recall what is meant by the induced measure space (A, F|A, |A).Why is it important to have A F. Show that the restrictionoff to A, f|A: (A, F|A) [0, +] is measurable.

2. We define the mapA :F [0, +] byA(E) =(A E), forall E F. Show that (, F, A) is a measure space.

3. Consider the equalities:

(f1A)d=

f d

A

=

(f|A)d|A (2)

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For each of the above integrals, what is the underlying measurespace on which the integral is considered. What is the map

being integrated. Explain why each integral is well defined.4. Show that in order to prove (2), it is sufficient to consider the

case when f is a simple function on (, F).

5. Show that in order to prove (2), it is sufficient to consider the

case when f is of the form f= 1B, for some B F.

6. Show that (2) is indeed true.

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Definition 45 Letf : (, F) [0, +] be a non-negative and mea-surable map, where(, F, ) is a measure space. letA F. We call

partial Lebesgue integraloffwith respect to overA, the integraldenotedA

f d, defined as:A

f d=

(f1A)d=

f dA =

(f|A)d|A

where

A

is the measure on(, F),

A

=(A ), f|A is the restric-tion of f to A and|A is the restriction of to F|A, the trace ofFonA.

Exercise 19.Let f, g : (, F) [0, +] be two non-negative andmeasurable maps. Let :

F [0, +

] be defined by (A) =

Af d,

for all A F.

1. Show thatis a measure on F.

2. Show that:

gd=

gfd

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Theorem 21 Letf : (, F)[0, +] be a non-negative and mea-surable map, where(, F, ) is a measure space. Let:F [0, +]be defined by(A) =

A

f d, for allA F. Then, is a measure onF, and for allg : (, F)[0, +] non-negative and measurable, wehave:

gd=

gfd

Definition 46 TheL1-spaces on a measure space(, F, ), are:

L1R(, F, )=f: (, F) (R, B(R)) measurable, |f|d


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sg g

1. Explain why the integral

|f|d makes sense.

2. Show thatf : (, F) (R, B(R)) is measurable, iff()R.

3. Show thatL1R

(, F, ) L1C

(, F, ).

4. Show thatL1R

(, F, ) ={fL1C

(, F, ) , f() R}

5. Show thatL1R

(, F, ) is closed under R-linear combinations.

6. Show thatL1C

(, F, ) is closed under C-linear combinations.

Definition 47 Letu: R be a real-valued function defined on a

set. We callpositive part andnegative part ofu the mapsu+andu respectively, defined asu+ = max(u, 0) andu = max(u, 0).

Exercise 21. Let fL1C

(, F, ). Let u= Re(f) and v =I m(f).

1. Show thatu = u+ u,v =v+ v,f=u+ u+ i(v+ v).

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g g

2. Show that|u| =u+ +u, |v|= v+ +v

3. Show thatu+, u, v+, v, |f|,u ,v, |u|, |v| all lie in L1R

(, F, ).

4. Explain why the integrals

u+d,

ud,

v+d,

vd areall well defined.

5. We define the integral offwith respect to , denoted

f d, as

f d=

u

+

d

u

d + i

v+

d

v

d

. Explain whyf dis a well defined complex number.6. In the case when f() C [0, +] = R+, explain why this

new definition of the integral offwith respect tois consistentwith the one already known (43) for non-negative and measur-able maps.

7. Show that

f d=

ud+i

vd and explain why all integralsinvolved are well defined.

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g g

Definition 48 Let f = u+ iv L1C

(, F, ) where (, F, ) is ameasure space. We define theLebesgue integral off with respectto

, denoted

f d, as:

f d=

u+d

ud+i

v+d

vd

Exercise 22.

Let f=u+iv L

1

C(, F, ) andA F.1. Show thatf1AL

1C

(, F, ).

2. Show thatfL1C

(, F, A).

3. Show thatf|AL1

C(A, F|A, |A)4. Show that

(f1A)d=

f dA =

f|Ad|A.

5. Show that 4. is:A

u+d A

ud+i

Av+d

A

vd

.

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Definition 49 Letf L1C

(, F, ) , where (, F, ) is a measurespace. let A F. We call partial Lebesgue integral of f withrespect to

over

A, the integral denoted

Af d, defined as:

A

f d=

(f1A)d=

f dA =

(f|A)d|A

whereA is the measure on(, F), A =(A ), f|A is the restric-

tion of f to A and|A is the restriction of to F|A, the trace ofFonA.

Exercise 23. Let f, gL1R

(, F, ) and let h= f+g

1. Show that: h+d+

fd+

gd=

hd+

f+d+

g+d

2. Show that

hd=

f d+

gd.

3. Show that

(f)d=

f d

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4. Show that if R then

(f)d=

f d.

5. Show that iffg then f d gd6. Show the following theorem.

Theorem 22 For allf, gL1C

(, F, ) and C, we have: (f+g)d=

f d+

gd

Exercise 24. Let f, g be two maps, and (fn)n1 be a sequence ofmeasurable maps fn: (, F) (C, B(C)), such that:

(i) , limn+

fn() =f() in C

(ii) n1 , |fn| g

(iii) gL1R(, F, )

Let (un)n1 be an arbitrary sequence in R.

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1. Show thatfL1C

(, F, ) and fnL1C

(, F, ) for all n 1.

2. For n 1, define hn = 2g |fn f|. Explain why Fatou

lemma (20) can be applied to the sequence (hn)n1.

3. Show that lim inf(un) = lim sup un.

4. Show that if R, then lim inf(+un) =+ lim infun.

5. Show thatun0 as n + if and only if lim sup |un|= 0.

6. Show that

(2g)d

(2g)d lim sup

|fn f|d

7. Show that lim sup

|fn f|d= 0.

8. Conclude that

|fn f|d 0 as n +.

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Theorem 23 (Dominated Convergence) Let (fn)n1 be a se-quence of measurable mapsfn: (, F) (C, B(C))such thatfnfin C2 . Suppose that there exists some g L1

R

(, F, ) such that|fn| g for alln 1. Thenf, fnL

1C

(, F, ) for alln 1, and:

limn+

|fn f|d= 0

Exercise 25.Let f L1C(, F, ) and put z =

f d. Let C,be such that ||= 1 andz =|z|. Put u=Re(f).

1. Show thatu L1R

(, F, )

2. Show thatu |f|3. Show that|

f d|=

(f)d.

4. Show that

(f)d=

ud.

2

i.e. for all

, the sequence (fn())n1 converges to f() C

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5. Prove the following theorem.

Theorem 24 Let f L1C

(, F, ) where (, F, ) is a measurespace. We have:

f d

|f|d

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Lebesgue Integral