Tutorial 5: Lebesgue Integration 1

5. Lebesgue IntegrationIn the following, (,F , ) is a measure space.Definition 39 Let A . We call characteristic function of A,the map 1A : R, defined by:

, 1A() 4={

1 if A0 if 6 A

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

where n 1, i R+ and Ai F , for all i = 1, . . . , n.

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Tutorial 5: Lebesgue Integration 2

Exercise 2. Show that s : (,F) (R+,B(R+)) is measurable,whenever s is a simple function on (,F).Exercise 3. Let s be a simple function on (,F) with representations =

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

() = (1A1(), . . . , 1An()). For each y s(), pick one y such that y = 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 of R+.

2. Show that s =

s() 1{s=}

3. Show that any simple function s can be represented as:

s =ni=1

i1Ai

where n 1, i R+, Ai F and = A1 ] . . . ]An.

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Tutorial 5: Lebesgue Integration 3

Definition 41 Let (,F) be a measurable space, and s be a simplefunction on (,F). We call partition of the simple function s, anyrepresentation of the form:

s =ni=1

i1Ai

where n 1, i R+, Ai F and = A1 ] . . . ]An.

Exercise 4. Let s be a simple function on (,F) with two partitions:

s =ni=1

i1Ai =mj=1

j1Bj

1. Show that s =

i,j i1AiBj is a partition of s.

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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Tutorial 5: Lebesgue Integration 4

3. Show thatni=1 i(Ai) =

mj=1 j(Bj).

4. Explain why the following definition is legitimate.

Definition 42 Let (,F , ) be a measure space, and s be a simplefunction on (,F). We define the integral of s with respect to , asthe sum, denoted I(s), defined by:

I(s) 4=ni=1

i(Ai) [0,+]

where s =n

i=1 i1Ai is any partition of s.

Exercise 5. Let s, t be two simple functions on (,F) with partitionss =

ni=1 i1Ai and t =

mj=1 j1Bj . Let R+.

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

s+ t =ni=1

mj=1

(i + j)1AiBj

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Tutorial 5: Lebesgue Integration 5

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

3. Show that s is a simple function on (,F).4. Show that I(s) = I(s).

5. Why is the notation I(s) meaningless if = + or < 0.6. Show that if s t then I(s) I(t).

Exercise 6. Let f : (,F) [0,+] be a non-negative and mea-surable map. For all n 1, we define:

sn4=

n2n1k=0

k

2n1{ k2nf< k+12n } + n1{nf} (1)

1. Show that sn 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 that sn sn+1 f , for all n 1.

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Tutorial 5: Lebesgue Integration 6

4. Show that sn f as n +1.

Theorem 18 Let f : (,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 that sn f .

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

fd, as:

fd4= sup{I(s) : s simple function on (,F) , s f}

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

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

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Tutorial 5: Lebesgue Integration 7

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

1. Show thatfd [0,+].

2. Show thatfd = I(f), whenever f is a simple function.

3. Show thatgd fd, whenever g : (,F) [0,+] is

non-negative and measurable map with g f .4. Show that

(cf)d = c

fd, if 0 < c < +. Explain why

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

6. Show thatfd = 0 ({f > 0}) = 0.

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Tutorial 5: Lebesgue Integration 8

7. Show that if s is a simple function on (,F) with s f , then({f > 0}) = 0 implies I(s) = 0.

8. Show thatfd = 0 ({f > 0}) = 0.

9. Show that

(+)fd = (+) fd. Explain why both inte-grals are well defined.

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

11. Show thatfd < + ({f = +}) = 0.

12. Suppose that () = + and take f = 1. Show that theconverse of the previous implication is not true.

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Tutorial 5: Lebesgue Integration 9

Exercise 8. Let s be a simple function on (,F). Let A F .1. Show that s1A is a simple function on (,F).2. Show that for any partition s =

ni=1 i1Ai of s, we have:

I(s1A) =ni=1

i(Ai A)

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

4. Suppose An F , An A. 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 fn f .

1. Recall what the notation fn f means.2. Explain why f : (,F) (R,B(R)) is measurable.

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Tutorial 5: Lebesgue Integration 10

3. Let = supn1fnd. Show that

fnd .

4. Show that fd.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 Fand An .

6. Show that cI(s1An) fnd, for all n 1.

7. Show that cI(s) .8. Show that I(s) .9. Show that

fd .

10. Conclude thatfnd

fd.

Theorem 19 (Monotone Convergence) Let (,F , ) be a mea-sure space. Let (fn)n1 be a sequence of non-negative and measurablemaps fn : (,F) [0,+] such that fn f . Then

fnd

fd.

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Tutorial 5: Lebesgue Integration 11

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

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 + gn f + g.

2. Show that

(f + g)d =fd+

gd.

3. Show that

(af + bg)d = afd+ b

gd.

Exercise 11. Let (fn)n1 be a sequence of non-negative and mea-surable maps fn : (,F) [0,+]. Define f =

+n=1 fn.

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

2. Show thatfd =

+n=1

fnd.

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Tutorial 5: Lebesgue Integration 12

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 write P() -a.s., if and only if:

N F , (N) = 0 , N c,P() holds

Exercise 12. Let P() be a property depending on , such that{ : P() holds} is an element of the -algebra F .

1. Show that P() , -a.s. ({ : P() holds}c) = 0.2. Explain why in general, the right-hand side of this equivalence

cannot be used to defined -almost sure properties.

Exercise 13. Let (,F , ) be a measure space and (An)n1 be asequence of elements of F . Show that (+n=1An)

+n=1 (An).

Exercise 14. Let (fn)n1 be a sequence of maps fn : [0,+].1. Translate formally the statement fn f -a.s.

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Tutorial 5: Lebesgue Integration 13

2. Translate formally fn f -a.s. and n, (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.. Let N F , (N) = 0 such thatf = g on N c. Explain why

fd =

(f1N)d +

(f1Nc)d, all

integrals being well defined. Show thatfd =

gd.

Exercise 16. Suppose (fn)n1 is a sequence of non-negative andmeasurable maps and f is a non-negative and measurable map, suchthat fn f -a.s.. Let N F , (N) = 0, such that fn f on N c.Define fn = fn1Nc and f = f1Nc .

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

2. Show that fn f .3. Show that

fnd

fd.

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Tutorial 5: Lebesgue Integration 14

Exercise 17. Let (fn)n1 be a sequence of non-negative and measur-able maps fn : (,F) [0,+]. For n 1, we define gn = infkn fk.

1. Explain why the gns are non-negative and measurable.

2. Show that gn lim inf fn.3. Show that

gnd

fnd, for all n 1.

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

5. Show that

(lim inf fn)d lim inffnd, and recall why all

integrals are well defined.

Theorem 20 (Fatou Lemma) Let (,F , ) be a measure space,and let (fn)n1 be a sequence of non-negative and measurable mapsfn : (,F) [0,+]. Then:

(lim infn+ fn)d lim infn+

fnd

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Tutorial 5: Lebesgue Integration 15

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 restrictionof f to A, f|A : (A,F|A) [0,+] is measurable.

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

3. Consider the equalities:(f1A)d =

fdA =

(f|A)d|A (2)

For each of the above integrals, what is the underlying measurespace on which the integral is considered. What is the mapbeing integrated. Explain why each integral is well defined.

4. Show that in order to prove (2), it is sufficient to consider thecase when f is a simple function on (,F).

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Tutorial 5: Lebesgue Integration 16

5. Show that in order to prove (2), it is sufficient to consider thecase when f is of the form f = 1B, for some B F .

6. Show that (2) is indeed true.

Definition 45 Let f : (,F) [0,+] be a non-negative and mea-surable map, where (,F , ) is a measure space. let A F . We callpartial Lebesgue integral of f with respect to over A, the integraldenoted

Afd, defined as:A

fd4=

(f1A)d =fdA =

(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 of Fon A.

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Tutorial 5: Lebesgue Integration 17

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

Afd,

for all A F .1. Show that is a measure on F .2. Show that:

gd =gfd

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

Afd, for all A F . Then, is a measure on

F , and for all g : (,F) [0,+] non-negative and measurable, wehave:

gd =gfd

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Tutorial 5: Lebesgue Integration 18

Definition 46 The L1-spaces on a measure space (,F , ), are:

L1R(,F , )4={f : (,F) (R,B(R)) measurable,

|f |d < +

}L1C(,F , )4=

{f : (,F) (C,B(C)) measurable,

|f |d < +

}

Exercise 20. Let f : (,F) (C,B(C)) be a measurable map.1. Explain why the integral

|f |d makes sense.2. Show that f : (,F) (R,B(R)) is measurable, if f() R.3. Show that L1R(,F , ) L1C(,F , ).4. Show that L1R(,F , ) = {f L1C(,F , ) , f() R}5. Show that L1R(,F , ) is closed under R-linear combinations.6. Show that L1C(,F , ) is closed under C-linear combinations.

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Tutorial 5: Lebesgue Integration 19

Definition 47 Let u : R be a real-valued function defined on aset . We call positive part and negative part of u the maps u+

and u respectively, defined as u+ = max(u, 0) and u = max(u, 0).

Exercise 21. Let f L1C(,F , ). Let u = Re(f) and v = Im(f).1. Show that u = u+u, v = v+v, f = u+u+ i(v+v).2. Show that |u| = u+ + u, |v| = v+ + v

3. Show that u+, u, v+, v, |f |, u, v, |u|, |v| all lie in L1R(,F , ).4. Explain why the integrals

u+d,

ud,

v+d,

vd are

all well defined.

5. We define the integral of f with respect to , denotedfd, as

fd =u+d ud+ i ( v+d vd). Explain why

fd is a well defined complex number.

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Tutorial 5: Lebesgue Integration 20

6. In the case when f() C [0,+] = R+, explain why thisnew definition of the integral of f with respect to is consistentwith the one already known (43) for non-negative and measur-able maps.

7. Show thatfd =

ud+i

vd and explain why all integrals

involved are well defined.

Definition 48 Let f = u + iv L1C(,F , ) where (,F , ) is ameasure space. We define the Lebesgue integral of f with respectto , denoted

fd, as:

fd4=u+d

ud+ i

(v+d

vd

)

Exercise 22. Let f = u+ iv L1C(,F , ) and A F .1. Show that f1A L1C(,F , ).

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Tutorial 5: Lebesgue Integration 21

2. Show that f L1C(,F , A).3. Show that f|A L1C(A,F|A, |A)4. Show that

(f1A)d =

fdA =

f|Ad|A.

5. Show that 4. is:A u

+d A ud+ i (A v+d A vd).Definition 49 Let f L1C(,F , ) , where (,F , ) is a measurespace. let A F . We call partial Lebesgue integral of f withrespect to over A, the integral denoted

A fd, defined as:

A

fd4=

(f1A)d =fdA =

(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 of Fon A.

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Tutorial 5: Lebesgue Integration 22

Exercise 23. Let f, g L1R(,F , ) and let h = f + g1. Show that:

h+d+fd+

gd =

hd+

f+d+

g+d

2. Show thathd =

fd+

gd.

3. Show that

(f)d = fd4. Show that if R then (f)d = fd.5. Show that if f g then fd gd6. Show the following theorem.

Theorem 22 For all f, g L1C(,F , ) and C, we have:(f + g)d =

fd+

gd

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Tutorial 5: Lebesgue Integration 23

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) n 1 , |fn| g(iii) g L1R(,F , )

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

1. Show that f L1C(,F , ) and fn L1C(,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 supun.4. Show that if R, then lim inf(+ un) = + lim inf un.5. Show that un 0 as n + if and only if lim sup |un| = 0.6. Show that

(2g)d (2g)d lim sup |fn f |d

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Tutorial 5: Lebesgue Integration 24

7. Show that lim sup |fn f |d = 0.

8. Conclude that |fn f |d 0 as n +.

Theorem 23 (Dominated Convergence) Let (fn)n1 be a se-quence of measurable maps fn : (,F) (C,B(C)) such that fn fin C2 . Suppose that there exists some g L1R(,F , ) such that|fn| g for all n 1. Then f, fn L1C(,F , ) for all n 1, and:

limn+

|fn f |d = 0

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

be such that || = 1 and z = |z|. Put u = Re(f).1. Show that u L1R(,F , )2. Show that u |f |2i.e. for all , the sequence (fn())n1 converges to f() C

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Tutorial 5: Lebesgue Integration 25

3. Show that | fd| = (f)d.4. Show that

(f)d =

ud.

5. Prove the following theorem.

Theorem 24 Let f L1C(,F , ) where (,F , ) is a measurespace. We have: fd |f |d

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5 Lebesgue Integration