tutorials in probability

7/31/2019 Tutorials in Probability

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A b d e l k a d e r BENHA RI

Tu t o r i a l s i n pr o b a b i l i t y


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Tutorial 1: Dynkin systems

1. Dynkin systems

Definition 1 A dynkin system on a set is a subset D of power set P(), with the following properties:

(i) D(ii) A,B D, A B B \ A D

(iii) An D, An An+1, n 1 +

n=1

An D

Definition 2 A -algebra on a set is a subset F of the pow

set P() with the following properties:(i) F

(ii) A F Ac= \ A F

(iii) An F, n 1 +

n=1

An F


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Exercise 1. Let F be a -algebra on . Show that F, thif A,B F then A B F and also A B F. Recall tB \ A = B Ac and conclude that F is also a dynkin system on

Exercise 2. Let (Di)iI be an arbitrary family of dynkin syste

on , with I= . Show that D = iI Di is also a dynkin system.

Exercise 3. Let (Fi)iI be an arbitrary family of-algebras on

with I= . Show that F= iI Fi is also a -algebra on .

Exercise 4. Let A be a subset of the power set P(). Define:

D(A)= {D dynkin system on : A D}

Show that P() is a dynkin system on , and that D(A) is not empDefine:

D(A)=

DD(A)

D


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Show that D(A) is a dynkin system on such that A D(A), athat it is the smallest dynkin system on with such property, (i.eD is a dynkin system on with A D, then D(A) D).

Definition 3 Let A P(). We call dynkin system generatbyA, the dynkin system on , denoted D(A), equal to the intersectof all dynkin systems on , which contain A.

Exercise 5. Do exactly as before, but replacing dynkin systems-algebras.

Definition 4 Let A P(). We call -algebra generated

A, the -algebra on , denoted (A), equal to the intersection of-algebras on , which contain A.

Definition 5 A subsetA of the power set P() is called a-systeon , if and only if it is closed under finite intersection, i.e. if it the property:

A,B A A B A


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Exercise 6. Let A be a -system on . For all A D(A), we defi

(A)= {B D(A) : A B D(A)}

1. IfA A, show that A (A)

2. Show that for all A D(A), (A) is a dynkin system on .

3. Show that ifA A, then D(A) (A).

4. Show that ifB D(A), then A (B).

5. Show that for all B D(A), D(A) (B).

6. Conclude that D(A) is also a -system on .

Exercise 7. Let D be a dynkin system on which is also a -syste

1. Show that ifA,B D then A B D.


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2. Let An D, n 1. Consider Bn= n

i=1 Ai. Show t+n=1An =

+n=1Bn.

3. Show that D is a -algebra on .

Exercise 8. Let A be a -system on . Explain why D(A) i-algebra on , and (A) is a dynkin system on . Conclude tD(A) = (A). Prove the theorem:

Theorem 1 (dynkin system) Let C be a collection of subsets owhich is closed under pairwise intersection. If D is a dynkin syst

containing C then D also contains the -algebra (C) generated by


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Tutorial 2: Caratheodorys Extension

Definition 7 A ring on is a subset R of the power set P() wthe following properties:

(i) R

(ii) A, B R A B R

(iii) A, B R A \ B R

Exercise 1. Show that A B = A \ (A \ B) and therefore tharing is closed under pairwise intersection.

Exercise 2.Show that a ring on is also a semi-ring on .

Exercise 3.Suppose that a set can be decomposed as = A

A2 A3 where A1, A2 and A3 are distinct from and . DefiS1

= {, A1, A2, A3, } and S2

= {, A1, A2 A3, }. Show that

and S2 are semi-rings on , but that S1 S2 fails to be a semi-ron .

Exercise 4. Let (Ri)iI be an arbitrary family of rings on , w

I = . Show that R= iI Ri is also a ring on .


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Exercise 5. Let A be a subset of the power set P(). Define:

R(A)= {R ring on : A R}

Show that P() is a ring on , and that R(A) is not empty. Defi

R(A)=

RR(A)

R

Show that R(A) is a ring on such that A R(A), and that ithe smallest ring on with such property, (i.e. if R is a ring onand A R then R(A) R).

Definition 8 Let A P(). We call ring generated by A, ring on , denoted R(A), equal to the intersection of all rings onwhich contain A.

Exercise 6.Let S be a semi-ring on . Define the set R of all finunions of pairwise disjoint elements of S, i.e.

R= {A : A = ni=1Ai for some n 0, Ai S}


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(where if n = 0, the corresponding union is empty, i.e. R). A = ni=1Ai and B =

pj=1Bj R:

1. Show that A B = i,j(Ai Bj) and that R is closed unpairwise intersection.

2. Show that if p 1 then A \ B = pj=1(ni=1(Ai \ Bj)).

3. Show that R is closed under pairwise difference.

4. Show that A B = (A \ B) B and conclude that R is a ron .

5. Show that R(S) = R.


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Exercise 7. Everything being as before, define:

R= {A : A = ni=1Ai for some n 0, Ai S}

(We do not require the sets involved in the union to be pairwise d

joint). Using the fact that R is closed under finite union, show thR R, and conclude that R = R = R(S).

Definition 9 Let A P() with A. We call measure onany map : A [0, +] with the following properties:

(i) () = 0

(ii) A A, An A and A =+n=1

An (A) =+n=1

(An)

The indicates that we assume the Ans to be pairwise disjointthe l.h.s. of (ii). It is customary to say in view of condition (ii) tha measure is countably additive.


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Exercise 8.If A is a -algebra on explain why property (ii) cbe replaced by:

(ii) An A and A =+

n=1

An (A) =+

n=1

(An)

Exercise 9. Let A P() with A and : A [0, +] bmeasure on A.

1. Show that ifA1, . . . , An A are pairwise disjoint and the unA = ni=1Ai lies in A, then (A) = (A1) + . . . + (An).

2. Show that ifA, B A, A B and B \A A then (A) (B

Exercise 10. Let S be a semi-ring on , and : S [0, +] bmeasure on S. Suppose that there exists an extension of on R(i.e. a measure : R(S) [0, +] such that |S = .


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1. Let A be an element of R(S) with representation A = ni=1as a finite union of pairwise disjoint elements of S. Show t(A) =

ni=1 (Ai)

2. Show that if : R(S) [0, +] is another measure w|S = , i.e. another extension of on R(S), then

= .

Exercise 11. Let S be a semi-ring on and : S [0, +] bmeasure. Let A be an element of R(S) with two representations:

A =n

i=1

Ai =

p

j=1

Bj

as a finite union of pairwise disjoint elements of S.

1. For i = 1, . . . , n, show that (Ai) =p

j=1 (Ai Bj)

2. Show thatn

i=1 (Ai) =p

j=1 (Bj)


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3. Explain why we can define a map : R(S) [0, +] as:

(A)=

ni=1

(Ai)

4. Show that () = 0.

Exercise 12. Everything being as before, suppose that (An)na sequence of pairwise disjoint elements of R(S), each An having representation:

An =

pn

k=1

Akn , n 1

as a finite union of disjoint elements of S. Suppose moreover tA = +n=1An is an element of R(S) with representation A =

pj=1

as a finite union of pairwise disjoint elements of S.

1. Show that for j = 1, . . . , p, Bj = +n=1

pnk=1 (A

kn Bj) a

explain why Bj is of the form Bj = +m=1Cm for some sequen

(Cm)m1 of pairwise disjoint elements of S.


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2. Show that (Bj) =+

n=1

pnk=1 (A

kn Bj)

3. Show that for n 1 and k = 1, . . . , pn, Akn = pj=1(A

kn Bj)

4. Show that (Akn) =pj=1 (Akn Bj)5. Recall the definition of of exercise (11) and show that it i

measure on R(S).

Exercise 13.Prove the following theorem:

Theorem 2 Let S be a semi-ring on . Let : S [0, +] b

measure on S. There exists a unique measure : R(S) [0, +such that |S = .


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Definition 10 We define an outer-measure on as being amap : P() [0, +] with the following properties:

(i) () = 0

(ii) A B (A) (B)

(iii)

+n=1

An

+n=1

(An)

Exercise 14. Show that (A B) (A) + (B), where

an outer-measure on and A, B .

Definition 11 Let be an outer-measure on . We define:

()= {A : (T) = (T A) + (T Ac) , T }

We call () the -algebra associated with the outer-measure

Note that the fact that () is indeed a -algebra on , remainsbe proved. This will be your task in the following exercises.


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Exercise 15. Let be an outer-measure on . Let = ()the -algebra associated with . Let A, B and T

1. Show that and Ac .

2. Show that (T A) = (T A B) + (T A Bc)

3. Show that T Ac = T (A B)c Ac

4. Show that T A Bc = T (A B)c A

5. Show that (T Ac) + (T A Bc) = (T (A B)c)

6. Adding

(T(AB)) on both sides 5., conclude that AB 7. Show that A B and A \ B belong to .

Exercise 16. Everything being as before, let An , n 1. DefiB1 = A1 and Bn+1 = An+1 \ (A1 . . . An). Show that the Bns pairwise disjoint elements of and that +n=1An =

+n=1Bn.


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Exercise 17. Everything being as before, show that ifB, C aB C = , then (T (B C)) = (T B) + (T C) for aT .

Exercise 18.Everything being as before, let (Bn)n1 be a sequen

of pairwise disjoint elements of , and let B = +n=1 Bn. Let N

1. Explain why Nn=1Bn

2. Show that (T (Nn=1Bn)) =N

n=1 (T Bn)

3. Show that (T Bc) (T (Nn=1Bn)c)

4. Show that (T Bc) +

+n=1

(T Bn) (T), and:

5. (T) (TBc)+(TB) (TBc)++

n=1 (TB

6. Show that B and (B) =+

n=1 (Bn).

7. Show that is a -algebra on , and | is a measure on


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Theorem 3 Let : P() [0, +] be an outer-measure onThen (), the so-called -algebra associated with , is indee-algebra on and |(), is a measure on (

).

Exercise 19. Let R be a ring on and : R [0, +] bmeasure on R. For all T , define:

(T)= inf

+n=1

(An) , (An) is an R-cover of T

where an R-cover ofT is defined as any sequence (An)n1 of eleme

of R such that T +n=1An. By convention inf

= + .

1. Show that () = 0.

2. Show that if A B then (A) (B).

3. Let (An)n1 be a sequence of subsets of , with (An) < +

for all n 1. Given > 0, show that for all n 1, there exi


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an R-cover (Apn)p1 of An such that:

+p=1

(Apn) < (An) + /2

n

Why is it important to assume (An) < +.

4. Show that there exists an R-cover (Rk) of +n=1An such tha

+k=1

(Rk) =

+n=1

+p=1

(Apn)

5. Show that (+n=1An) +

+n=1 (An)

6. Show that is an outer-measure on .


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Exercise 20. Everything being as before, Let A R. Let (An)nbe an R-cover of A and put B1 = A1 A, and:

Bn+1= (An+1 A) \ ((A1 A) . . . (An A))

1. Show that (A) (A).

2. Show that (Bn)n1 is a sequence of pairwise disjoint elemeof R such that A = +n=1Bn.

3. Show that (A) (A) and conclude that |R = .

Exercise 21.Everything being as before, Let A R and T

1. Show that (T) (T A) + (T Ac).

2. Let (Tn) be an R-cover ofT. Show that (Tn A) and (Tn Aare R-covers of T A and T Ac respectively.

3. Show that (T A) + (T Ac) (T).


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4. Show that R ().

5. Conclude that (R) ().

Exercise 22.

Prove the following theorem:

Theorem 4 (caratheodorys extension) Let R be a ring onand : R [0, +] be a measure on R. There exists a meas : (R) [0, +] such that |R = .

Exercise 23. Let Sbe a semi-ring on . Show that (R(S)) = (

Exercise 24.Prove the following theorem:

Theorem 5 Let S be a semi-ring on and : S [0, +] bmeasure on S. There exists a measure : (S) [0, +] such t|S = .


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Tutorial 3: Stieltjes-Lebesgue Measure

3. Stieltjes-Lebesgue Measure

Definition 12 Let A P() and : A [0, +] be a map. say that is finitely additive if and only if, given n 1:

A A, Ai A, A =ni=1

Ai (A) =ni=1

(Ai)

We say that is finitely sub-additive if and only if, given n

A A, Ai A, A ni=1

Ai (A) ni=1

(Ai)

Exercise 1. Let S= {]a, b] , a , b R} be the set of all interv

]a, b], defined as ]a, b] = {x R, a < x b}.

1. Show that ]a, b]]c, d] =]a c, b d]

2. Show that ]a, b]\]c, d] =]a, b c]]a d, b]


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3. Show that c d b c a d.

4. Show that S is a semi-ring on R.

Exercise 2. Suppose S is a semi-ring in and : S [0, +finitely additive. Show that can be extended to a finitely additmap : R(S) [0, +], with |S = .

Exercise 3. Everything being as before, Let A R(S), Ai R(A ni=1Ai where n 1. Define B1 = A1A and for i = 1, . . . , n

Bi+1= (Ai+1 A) \ ((A1 A) . . . (Ai A))

1. Show that B1, . . . , Bn are pairwise disjoint elements of Rsuch that A = ni=1Bi.

2. Show that for all i = 1, . . . , n, we have (Bi) (Ai).

3. Show that is finitely sub-additive.

4. Show that is finitely sub-additive.


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Exercise 4. Let F : R R be a right-continuous, non-decreasmap. Let S be the semi-ring on R, S = {]a, b] , a,b R}. Define map : S [0, +] by () = 0, and:

a b , (]a, b])

= F(b) F(a) Let a < b and ai < bi for i = 1, . . . , n and n 1, with :

]a, b] =n

i=1

]ai, bi]

1. Show that there is i1 {1, . . . , n} such that ai1 = a.

2. Show that ]bi1 , b] = i{1,...,n}\{i1}]ai, bi]

3. Show the existence of a permutation (i1, . . . , in) of {1, . . . ,such that a = ai1 < bi1 = ai2 < . . . < bin = b.

4. Show that is finitely additive and finitely sub-additive.


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Exercise 5. being defined as before, suppose a < b and an 0, show that there is ]0, b a[ such that:

0 F(a + ) F(a)

4. For n 1, show that there is n > 0 such that:

0 F(bn + n) F(bn)

2n


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5. Show that [a + , b] +n=1]an, bn + n[.

6. Explain why there exist p 1 and integers n1, . . . , np such th

]a + , b] pk=1]ank , bnk + nk ]

7. Show that F(b) F(a) 2 ++

n=1 F(bn) F(an)

8. Show that : S [0, +] is a measure.

Definition 13 A topology on is a subset T of the power P(), with the following properties:

(i) , T

(ii) A, B T A B T

(iii) Ai T, i I iI

Ai T


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Property (iii) of definition (13) can be translated as: for any fam(Ai)iI of elements of T, the union iIAi is still an element ofHence, a topology on , is a set of subsets of containing athe empty set, which is closed under finite intersection and arbitr

union.Definition 14 A topological space is an ordered pair (, T), wh is a set and T is a topology on .

Definition 15 Let (, T) be a topological space. We say that A is an open set in , if and only if it is an element of the topology

We say that A is a closed set in , if and only if its complemAc is an open set in .

Definition 16 Let (, T) be a topological space. We define borel -algebra on , denoted B(), as the -algebra on , genated by the topology T. In other words, B() = (T)


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Definition 17 We define the usual topology on R, denoted Tas the set of all U R such that:

x U , > 0 , ]x , x + [ U

Exercise 6.Show that TR is indeed a topology on R.

Exercise 7. Consider the semi-ring S= {]a, b] , a , b R}. Let

be the usual topology on R, and B(R) be the borel -algebra on

1. Let a b. Show that ]a, b] = +n=1]a, b + 1/n[.

2. Show that (S) B(R).3. Let U be an open subset of R. Show that for all x U, th

exist ax, bx Q such that x ]ax, bx] U.

4. Show that U = xU]ax, bx].

5. Show that the set I= {]ax, bx] , x U} is countable.


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6. Show that U can be written U = iIAi with Ai S.

7. Show that (S) = B(R).

Theorem 6 Let S be the semi-ring S = {]a, b] , a, b R}. Ththe borel -algebra B(R) on R, is generated by S, i.e. B(R) = (

Definition 18 A measurable space is an ordered pair (, F) wh is a set and F is a -algebra on .

Definition 19 A measure space is a triple (, F, ) where (,is a measurable space and : F [0, +] is a measure on F.


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Exercise 8.Let (, F, ) be a measure space. Let (An)n1 bsequence of elements ofF such that An An+1 for all n 1, andA = +n=1An (we write An A). Define B1 = A1 and for all n Bn+1 = An+1 \ An.

1. Show that (Bn) is a sequence of pairwise disjoint elements osuch that A = +n=1Bn.

2. Given N 1 show that AN = Nn=1Bn.

3. Show that (AN) (A) as N +

4. Show that (An) (An+1) for all n 1.

Theorem 7 Let (, F, ) be a measure space. Then if (An)n1 isequence of elements of F, such that An A, we have (An) (A

1i.e. the sequence ((An))n1 is non-decreasing and converges to (A).


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Exercise 9.Let (, F, ) be a measure space. Let (An)n1 bsequence of elements ofF such that An+1 An for all n 1, andA = +n=1An (we write An A). We assume that (A1) < +.

1. Define Bn

= A

1 \An

and show that Bn F

, Bn

A1 \

A.

2. Show that (Bn) (A1 \ A)

3. Show that (An) = (A1) (A1 \ An)

4. Show that (A) = (A1) (A1 \ A)

5. Why is (A1) < + important in deriving those equalities.

6. Show that (An) (A) as n +

7. Show that (An+1) (An) for all n 1.

Theorem 8 Let (, F, ) be a measure space. Then if (An)n1a sequence of elements of F, such that An A and (A1) < +, have (An) (A).


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Exercise 10.Take = R and F = B(R). Suppose is a meason B(R) such that (]a, b]) = b a, for a < b. Take An =]n, +[

1. Show that An .

2. Show that (An) = +, for all n 1.

3. Conclude that (An) () fails to be true.

Exercise 11. Let F : R R be a right-continuous, non-decreasmap. Show the existence of a measure : B(R) [0, +] such th

a, b R , a b , (]a, b]) = F(b) F(a)

Exercise 12.Let 1, 2 be two measures on B(R) with property (For n 1, we define:

Dn= {B B(R) , 1(B] n, n]) = 2(B] n, n])}

1. Show that Dn is a dynkin system on R.


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2. Explain why 1(] n, n]) < + and 2(] n, n]) < +needed when proving 1.

3. Show that S= {]a, b] , a , b R} Dn.

4. Show that B(R) Dn.

5. Show that 1 = 2.

6. Prove the following theorem.

Theorem 9 Let F : R R be a right-continuous, non-decreasmap. There exists a unique measure : B(R) [0, +] such tha

a, b R , a b , (]a, b]) = F(b) F(a)

Definition 20 LetF : R R be a right-continuous, non-decreasmap. We call stieltjes measure on R associated with F, the unimeasure on B(R), denoted dF, such that:

a, b R , a b , dF(]a, b]) = F(b) F(a)


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Definition 21 We call lebesgue measure on R, the unique m

sure on B(R), denoted dx, such that:

a, b R , a b , dx(]a, b]) = b a

Exercise 13. Let F : R R be a right-continuous, non-decreasmap. Let x0 R.

1. Show that the limit F(x0) = limx


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Exercise 14.Let F : R R be a right-continuous, non-decreasmap. Let a b.

1. Show that ]a, b] B(R) and dF(]a, b]) = F(b) F(a)

2. Show that [a, b] B(R) and dF([a, b]) = F(b) F(a)

3. Show that ]a, b[ B(R) and dF(]a, b[) = F(b) F(a)

4. Show that [a, b[ B(R) and dF([a, b[) = F(b) F(a)

Exercise 15. Let A be a subset of the power set P(). Let

Define: A|= {A , A A}

1. Show that if A is a topology on , A| is a topology on

2. Show that if A is a -algebra on , A| is a -algebra on


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Definition 22 Let be a set, and . Let A be a subsetthe power set P(). We call trace of A on , the subset A| of power set P() defined by:

A|=

{A

, A

A}

Definition 23 Let(, T) be a topological space and . We cinduced topology on , denoted T| , the topology on definby:

T|= {A , A T }

In other words, the induced topology T|

is the trace of T on .

Exercise 16.Let A be a subset of the power set P(). Let and A| be the trace of A on . Define:

= {A (A) , A (A| )}

where (A| ) refers to the -algebra generated by A| on .


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1. Explain why the notation (A| ) by itself is ambiguous.

2. Show that A .

3. Show that is a -algebra on .

4. Show that (A|) = (A)|

Theorem 10 Let and A be a subset of the power set P(Then, the trace on of the -algebra (A) generated by A, is eqto the -algebra on generated by the trace of A on . In otwords, (A)| = (A|).

Exercise 17.Let (, T) be a topological space and withinduced topology T| .

1. Show that B()| = B().

2. Show that if B() then B() B().


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3. Show that B(R+) = {A R+ , A B(R)}.

4. Show that B(R+) B(R).

Exercise 18.

Let (, F, ) be a measure space and

1. Show that (, F|) is a measurable space.

2. If F, show that F| F.

3. If F, show that (, F| , | ) is a measure space, wh| is defined as | = |(F|).

Exercise 19. Let F : R+ R be a right-continuous, non-decreasmap with F(0) 0. Define:

F(x)=

0 if x < 0F(x) if x 0

1. Show that F : R R is right-continuous and non-decreasin


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2. Show that : B(R+) [0, +] defined by = dF|B(R+), imeasure on B(R+) with the properties:

(i) ({0}) = F(0)

(ii) 0 a b , (]a, b]) = F(b) F(a)

Exercise 20. Define: C = {{0} } {]a, b] , 0 a b}

1. Show that C B(R+)

2. Let U be open in R+. Show that U is of the form:

U =iI

(R+

]ai, bi])

where I is a countable set and ai, bi R with ai bi.

3. For all i I, show that R+]ai, bi] (C).

4. Show that (C) = B(R+)


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Exercise 21.Let 1 and 2 be two measures on B(R+) with:

(i) 1({0}) = 2({0}) = F(0)

(ii) 1(]a, b]) = 2(]a, b]) = F(b) F(a)

for all 0 a b. For n 1, we define:

Dn = {B B(R+) , 1(B [0, n]) = 2(B [0, n])}

1. Show that Dn is a dynkin system on R+ with C Dn, wh

the set C is defined as in exercise (20).

2. Explain why 1([0, n]) < + and 2([0, n]) < + is import

when proving 1.

3. Show that 1 = 2.



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Theorem 11 Let F : R+ R be a right-continuous, non-decreasmap with F(0) 0. There exists a unique : B(R+) [0, +measure on B(R+) such that:

(i) ({0}) = F(0)

(ii) 0 a b , (]a, b]) = F(b) F(a)

Definition 24 LetF : R+ R be a right-continuous, non-decreasmap with F(0) 0. We call stieltjes measure on R+ associawith F, the unique measure on B(R+), denoted dF, such that:

(i) dF({0}) = F(0)(ii) 0 a b , dF(]a, b]) = F(b) F(a)


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Tutorial 4: Measurability

4. Measurability

Definition 25 Let A and B be two sets, and f : A B be a mGiven A A, we call direct image of A by f the set denoted f(Aand defined by f(A) =

{f(x) : x

A

}.

Definition 26 Let A and B be two sets, and f : A B be a mGiven B B, we call inverse image of B by f the set denof1(B), and defined by f1(B) = {x : x A , f(x) B}.

Exercise 1. Let A and B be two sets, and f : A B be a bijectfrom A to B. Let A

A and B

B.

1. Explain why the notation f1(B) is potentially ambiguous

2. Show that the inverse image of B by f is in fact equal to tdirect image ofB by f1.

3. Show that the direct image of A by f is in fact equal to tinverse image of A by f1.


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Definition 27 Let (, T) and (S, TS) be two topological spaces.map f : S is said to be continuous if and only if:

B TS , f1(B) T

In other words, if and only if the inverse image of any open set inis an open set in .

We Write f : (, T) (S, TS) is continuous, as a way of emphasizthe two topologies T and TS with respect to which f is continuouDefinition 28 Let E be a set. A map d : E E [0, +[ is sto be a metric on E, if and only if:

(i) x, y E , d(x, y) = 0 x = y(ii) x, y E , d(x, y) = d(y, x)

(iii) x,y,z E , d(x, y) d(x, z) + d(z, y)

Definition 29 A metric space is an ordered pair (E, d) whereis a set, and d is a metric on E.


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Definition 30 Let (E, d) be a metric space. For all x E a > 0, we define the so-called open ball in E:

B(x, )= {y : y E , d(x, y) < }

We call metric topology on E, associated with d, the topologydefined by:

TdE = {U E , x U, > 0, B(x, ) U}

Exercise 2. Let TdE be the metric topology associated with d, wh(E, d) is a metric space.

1. Show that TdE is indeed a topology on E.2. Given x E and > 0, show that B(x, ) is an open set in

Exercise 3. Show that the usual topology on R is nothing but metric topology associated with d(x, y) = |x y|.


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Exercise 4. Let (E, d) and (F, ) be two metric spaces. Show ta map f : E F is continuous, if and only if for all x E and >there exists > 0 such that for all y E:

d(x, y) <

(f(x), f(y)) <

Definition 31 Let (, T) and (S, TS) be two topological spaces.map f : S is said to be a homeomorphism, if and only if fa continuous bijection, such that f1 is also continuous.

Definition 32 A topological space (,

T) is said to be metrizab

if and only if there exists a metric d on , such that the associametric topology coincides with T, i.e. Td = T.


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Definition 33 Let (E, d) be a metric space and F E. We induced metric on F, denoted d|F, the restriction of the metrito F F, i.e. d|F = d|FF.

Exercise 5.Let (E, d) be a metric space and F E. We defiTF = (TdE )|F as the topology on F induced by the metric topologyE. Let TF = T

d|FF be the metric topology on F associated with

induced metric d|F on F.

1. Show that TF TF.2. Given A

TF, show that A = (

xAB(x, x))

F for so

x > 0, x A, where B(x, x) denotes the open ball in E.3. Show that TF TF.

Theorem 12 Let (E, d) be a metric space and F E. Then, topology on F induced by the metric topology, is equal to the met

topology on F associated with the induced metric, i.e. (

TdE )|F =

T

d

F


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Exercise 6. Let : R ] 1, 1[ be the map defined by:x R , (x) = x|x| + 1

1. Show that [

1, 0[ is not open in R.

2. Show that [1, 0[ is open in [1, 1].3. Show that is a homeomorphism between R and ] 1, 1[.4. Show that limx+ (x) = 1 and limx (x) = 1.

Exercise 7. Let R = [, +] = R{, +}. Let be definas in exercise (6), and : R [1, 1] be the map defined by:

(x) =

(x) if x R1 if x = +

1 if x = Define:

TR = {U R , (U) is open in [1, 1]}


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1. Show that is a bijection from R to [1, 1], and let =

2. Show that TR is a topology on R.3. Show that is a homeomorphism between R and [

1, 1].

4. Show that [, 2[, ]3, +], ]3, +[ are open in R.5. Show that if : R [1, 1] is an arbitrary homeomorphis

then U R is open, if and only if (U) is open in [1, 1].

Definition 34 The usual topology on R is defined as:

TR = {U R , (U) is open in [1, 1]}where : R [1, 1] is defined by () = 1, (+) = 1 and

x R , (x) = x|x| + 1


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Exercise 8. Let and be as in exercise (7). Define:

T = (TR)|R = {U R , U TR}1. Recall why

T is a topology on R.

2. Show that for all U R, (U R) = (U)] 1, 1[.3. Explain why if U TR, (U R) is open in ] 1, 1[.4. Show that T TR, (the usual topology on R).5. Let U TR. Show that (U) is open in ] 1, 1[ and [1, 1].6. Show that TR TR7. Show that TR = T, i.e. that the usual topology on R indu

the usual topology on R.

8. Show that B(R) = B(R)|R = {B R , B B(R)}


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Exercise 9.Let d : R R [0, +[ be defined by:(x, y) R R , d(x, y) = |(x) (y)|

where is an arbitrary homeomorphism from R to [1, 1].1. Show that d is a metric on R.

2. Show that if U TR, then (U) is open in [1, 1]3. Show that for all U TR and y (U), there exists > 0 su

that:z [1, 1] , |z y| < z (U)

4. Show that TR TdR.5. Show that for all U Td

Rand x U, there is > 0 such tha

y R , |(x) (y)| < y U

6. Show that for all U TdR

, (U) is open in [1, 1].


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7. Show that TdR

TR8. Prove the following theorem.

Theorem 13 The topological space (R, TR) is metrizable.

Definition 35 Let (, F) and (S, ) be two measurable spaces.map f : S is said to be measurable with respect to Fand and only if:

B , f1(B) F

We Write f : (, F) (S, ) is measurable, as a way of emphasizthe two -algebra Fand with respect to which f is measurableExercise 10. Let (, F) and (S, ) be two measurable spaces. S be a set and f : S be a map such that f() S S. define as the trace of on S, i.e. = |S .

1. Show that for all B

, we have f1(B) = f1(B

S)


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2. Show that f : (, F) (S, ) is measurable, if and onlyf : (, F) (S, ) is itself measurable.

3. Let f : R+. Show that the following are equivalent:

(i) f : (, F) (R+

, B(R+

)) is measurable(ii) f : (, F) (R, B(R)) is measurable

(iii) f : (, F) (R, B(R)) is measurable

Exercise 11. Let (, F), (S, ), (S1, 1) be three measurable spaclet f : (, F) (S, ) and g : (S, ) (S1, 1) be two measuramaps.

1. For all B S1, show that (g f)1(B) = f1(g1(B))2. Show that g f : (, F) (S1, 1) is measurable.


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Exercise 12.Let (, F) and (S, ) be two measurable spaces. f : S be a map. We define:

= {B , f1(B) F}

1. Show that f1(S) = .

2. Show that for all B S, f1(Bc) = (f1(B))c.3. Show that ifBn S, n 1, then f1(+n=1Bn) = +n=1f1(B4. Show that is a -algebra on S.


Theorem 14 Let (, F) and (S, ) be two measurable spaces, aA be a set of subsets of S generating , i.e. such that = (AThen f : (, F) (S, ) is measurable, if and only if:

B A , f1(B) F


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Exercise 13. Let (, T) and (S, TS) be two topological spaces. Lf : S be a map. Show that if f : (, T) (S, TS) is continuothen f : (, B()) (S, B(S)) is measurable.Exercise 14.We define the following subsets of the power set

P(R

C1 = {[, c] , c R}C2 = {[, c[ , c R}C3 = {[c, +] , c R}C4 = {]c, +] , c R}

1. Show that C2 and C4 are subsets ofTR.2. Show that the elements ofC1 and C3 are closed in R.3. Show that for all i = 1, 2, 3, 4, (Ci) B(R).4. Let U be open in R. Explain why U R is open in R.


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5. Show that any open subset of R is a countable union of opbounded intervals in R.

6. Let a < b, a, b R. Show that we have:

]a, b[=

+n=1

]a, b 1/n] =+n=1

[a + 1/n,b[

7. Show that for all i = 1, 2, 3, 4, ]a, b[ (Ci).8. Show that for all i = 1, 2, 3, 4, {{}, {+}} (Ci).

9. Show that for all i = 1, 2, 3, 4, (Ci) = B(R)



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Theorem 15 Let (, F) be a measurable space, and f : Ra map. The following are equivalent:

(i) f : (, F) (R, B(R)) is measurable(ii)

B

B(R) ,

{f

B

} F(iii) c R , {f c} F(iv) c R , {f < c} F(v) c R , {c f} F

(vi) c R , {c < f} F

Exercise 15. Let (, F) be a measurable space. Let (fn)n1 bsequence of measurable maps fn : (, F) (R, B(R)). Let g and hthe maps defined by g() = infn1 fn() and h() = supn1 fn(for all .

1. Let c R. Show that {c g} = +n=1{c fn}.2. Let c

R. Show that

{h

c

}=

+n=1

{fn

c

}.


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3. Show that g, h : (, F) (R, B(R)) are measurable.

Definition 36 Let (vn)n1 be a sequence in R. We define:

u = liminfn+

vn = supn1

infkn

vk

and:

w= limsup

n+vn

= inf

n1

supkn

vk

Then, u, w R are respectively called lower limit and upper limof the sequence

(vn)n1.

Exercise 16. Let (vn)n1 be a sequence in R. for n 1 we defiun = infkn vk and wn = supkn vk. Let u and w be the lower limand upper limit of (vn)n1, respectively.

1. Show that un un+1 u, for all n 1.


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2. Show that w wn+1 wn, for all n 1.3. Show that un u and wn w as n +.4. Show that un

vn

wn, for all n

1.

5. Show that u w.6. Show that if u = w then (vn)n1 converges to a limit v

with u = v = w.

7. Show that if a, b R are such that u < a < b < w then forn

1, there exist N1, N2

n such that vN1 < a < b < vN2 .

8. Show that if a, b R are such that u < a < b < w then thexist two strictly increasing sequences of integers (nk)k1 a(mk)k1 such that for all k 1, we have vnk < a < b < vmk

9. Show that if (vn)n1 converges to some v R, then u = w.


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Theorem 16 Let (vn)n1 be a sequence in R. Then, the followare equivalent:

(i) lim inf n+

vn = lim supn+

vn

(ii) limn+ vn exists in R.

in which case:

limn+

vn = lim infn+

vn = lim supn+

vn

Exercise 17. Let f, g : (,

F)

(R,

B(R)) be two measura

maps, where (, F) is a measurable space.1. Show that {f < g} = rQ({f < r} {r < g}).2. Show that the sets {f < g}, {f > g}, {f = g}, {f g}, {f

belong to the -algebra F.


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Exercise 18. Let (, F) be a measurable space. Let (fn)n1a sequence of measurable maps fn : (, F) (R, B(R)). We defig = lim inffn and h = lim sup fn in the obvious way:

, g()

= liminf

n+fn

()

, h() = limsupn+

fn()

1. Show that g, h : (, F) (R, B(R)) are measurable.2. Show that g h, i.e. , g() h().3. Show that {g = h} F.4. Show that { : , limn+ fn() exists in R} F.5. Suppose = {g = h}, and let f() = limn+ fn(), for

. Show that f : (, F) (R, B(R)) is measurable.


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Exercise 19. Let f, g : (, F) (R, B(R)) be two measuramaps, where (, F) is a measurable space.

1. Show that f, |f|, f+ = max(f, 0) and f = max(f, 0) measurable with respect to

Fand

B(R).

2. Let a R. Explain why the map a + f may not be well defin3. Show that (a+f) : (, F) (R, B(R)) is measurable, whene

a R.4. Show that (a.f) : (, F) (R, B(R)) is measurable, for

a

R. (Recall the convention 0.

= 0).

5. Explain why the map f + g may not be well defined.

6. Suppose that f 0 and g 0, i.e. f() [0, +] and ag() [0, +]. Show that {f + g < c} = {f < c g}, forc R. Show that f + g : (, F) (R, B(R)) is measurable


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7. Show that f+ g : (, F) (R, B(R)) is measurable, in the cwhen f and g take values in R.

8. Show that 1/f : (, F) (R, B(R)) is measurable, in the cwhen f()

R

\ {0}

.

9. Suppose that f is R-valued. Show that f defined by f()f() if f() = 0 and f() = 1 if f() = 0, is measurable wrespect to Fand B(R).

10. Suppose f and g take values in R. Let f be defined as inShow that for all c R, the set {f g < c} can be expressed a({f > 0}{g < c/f})({f < 0}{g > c/f})({f = 0}{f <

11. Show that f g : (, F) (R, B(R)) is measurable, in the cwhen f and g take values in R.


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Exercise 20.Let f, g : (, F) (R, B(R)) be two measurable mawhere (, F) is a measurable space. Let f , g, be defined by:

f()=

f() if f() {, +}

1 if f()

{, +

}g() being defined in a similar way. Consider the partitions of = A1 A2 A3 A4 A5 and = B1 B2 B3 B4 Bwhere A1 = {f ]0, +[}, A2 = {f ] , 0[}, A3 = {f = A4 = {f = }, A5 = {f = +} and B1, B2, B3, B4, B5 bedefined in a similar way with g. Recall the conventions 0 (+) =() (+) = (), etc...

1. Show that f and g are measurable with respect to Fand B(R2. Show that all Ais and Bj s are elements ofF.3. Show that for all B B(R):

{f g B} =5

i,j=1

(Ai Bj {f g B})


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4. Show that Ai Bj {f g B} = Ai Bj {fg B}, in tcase when 1 i 3 and 1 j 3.

5. Show that Ai Bj {f g B} is either equal to or Ai in the case when i

4 or j

4.

6. Show that f g : (, F) (R, B(R)) is measurable.

Definition 37 Let (, T) be a topological space, and A . call closure of A in , denoted A, the set defined by:

A=

{x

: x

U

T U

A

=

}Exercise 21. Let (E, T) be a topological space, and A E. Letbe the closure of A.

1. Show that A A and that A is closed.2. Show that if B is closed and A

B, then A

B.


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3. Show that A is the smallest closed set in E containing A.

4. Show that A is closed if and only if A = A.

5. Show that if (E,

T) is metrizable, then:

A = {x E : > 0 , B(x, ) A = }where B(x, ) is relative to any metric d such that TdE = T.

Exercise 22. Let (E, d) be a metric space. Let A E. For x E, we define:

d(x, A)

= inf{d(x, y) : y A}

= A(x)where it is understood that inf = +.

1. Show that for all x E, d(x, A) = d(x, A).2. Show that d(x, A) = 0, if and only if x A.3. Show that for all x, y

E, d(x, A)

d(x, y) + d(y, A).


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4. Show that if A = , |d(x, A) d(y, A)| d(x, y).5. Show that A : (E, TdE ) (R, TR) is continuous.6. Show that if A is closed, then A = 1A (

{0

})

Exercise 23.Let (, F) be a measurable space. Let (fn)n1 bsequence of measurable maps fn : (, F) (E, B(E)), where (E, da metric space. We assume that for all , the sequence (fn())nconverges to some f() E.

1. Explain why lim inffn and lim sup fn may not be defined in

arbitrary metric space E.

2. Show that f : (, F) (E, B(E)) is measurable, if and onlyf1(A) Ffor all closed subsets A of E.

3. Show that for all A closed in E, f1(A) = (A f)1({0where the map A : E R is defined as in exercise (22).


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4. Show that A fn : (, F) (R, B(R)) is measurable.5. Show that f : (, F) (E, B(E)) is measurable.

Theorem 17 Let (,F

) be a measurable space. Let (fn)n1 b

sequence of measurable maps fn : (, F) (E, B(E)), where (Eis a metric space. Then, if the limit f = lim fn exists on , the mf : (, F) (E, B(E)) is itself measurable.

Definition 38 The usual topology on C, the set of complex nu

bers, is defined as the metric topology associated withd(z, z) = |z

Exercise 24. Let f : (, F) (C, B(C)) be a measurable mwhere (, F) is a measurable space. Let u = Re(f) and v = Im(Show that u,v, |f| : (, F) (R, B(R)) are all measurable.Exercise 25. Define the subset of the power set P(C):

C

=

{]a, b[

]c, d[ , a,b,c,d

R

}


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where it is understood that:

]a, b[]c, d[ = {z = x + iy C , (x, y) ]a, b[]c, d[}1. Show that any element of C is open in C.2. Show that (C) B(C).3. Let z = x + iy C. Show that if |x| < and |y| < then

have |z| < 2.4. Let U be open in C. Show that for all z U, there are ratio

numbers az, bz, cz, dz such that z ]az , bz[]cz, dz[ U.5. Show that U can be written as U = +n=1An where An C.6. Show that (C) = B(C).7. Let (, F) be a measurable space, and u, v : (, F) (R, B(R

be two measurable maps. Show that u+iv : (, F) (C, B(Cis measurable.


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Tutorial 5: Lebesgue Integration

5. Lebesgue Integration

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

Definition 39 Let A . We call characteristic function of

the map 1A : R, defined by:

, 1A()=

1 if A0 if A

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

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

s =ni=1

i1Ai

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


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

Exercise 3. Let s be a simple function on (, F) with representats = ni=1 i1Ai. Consider the map : {0, 1}n defined () = (1A1(), . . . , 1An()). For each y s(), pick one y such that y = s(y). Consider the map : s() {0, 1}

n defined(y) = (y).

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

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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3. Show thatn

i=1 i(Ai) =m

j=1 j(Bj).

4. Explain why the following definition is legitimate.

Definition 42 Let (, F, ) be a measure space, and s be a sim

function on (, F). We define the integral of s with respect to ,the sum, denoted I(s), defined by:

I(s)=

ni=1

i(Ai) [0, +]

where s =

ni=1 i1Ai is any partition of s.

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

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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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 msurable map. For all n 1, we define:

sn =n2n1k=0

k2n

1{ k2nf


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4. Show that sn f as n +1.

Theorem 18 Let f : (, F) [0, +] be a non-negative and msurable map, where (, F) is a measurable space. There exists a

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

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

f d, as:

f d

= sup{I

(s) : s simple function on (, F) , s f}

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

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


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

1. Show that

f d [0, +].

2. Show that

f d = I(f), whenever f is a simple function.

3. Show that

gd

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

4. Show that

(cf)d = c

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

5. For n 1, put An = {f > 1/n}, and sn = (1/n)1An. Shthat sn is a simple function on (, F) with sn f. Show tAn {f > 0}.

6. Show that

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


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7. Show that if s is a simple function on (, F) with s f, th({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 ingrals are well defined.

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

11. Show that

f d < + ({f = +}) = 0.

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


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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 th 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 measable 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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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.

c ]0, 1[. For n 1, define An = {cs fn}. Show that An and An .

6. Show that cI(s1An)

fnd, for all n 1.

7. Show that cI(s) .

8. Show that I(s) .

9. Show that

f d .

10. Conclude that

fnd

f d.

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


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

1. Show that if (fn)n1 and (gn)n1 are two sequences of nonegative and measurable maps such that fn f and gn then fn + gn f + 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 m

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

+n=1 fn.

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

2. Show that

f d =+

n=1

fnd.


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Definition 44 Let (, F, ) be a measure space and let P() bproperty depending on . We say that the property P() ho-almost surely, and we write P() -a.s., if and only if:

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

Exercise 12. Let P() be a property depending on , such t{ : 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 equivale

cannot be used to defined -almost sure properties.

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

sequence 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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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-negatand measurable with f = g -a.s.. Let N F, (N) = 0 such thf = g on Nc. Explain why

f d =

(f1N)d +

(f1Nc)d,

integrals being well defined. Show that

f d =

gd.

Exercise 16. Suppose (fn)n1 is a sequence of non-negative ameasurable maps such that fn f -a.s.. Let N F, (N) = 0, suthat fn f on Nc. Define fn = fn1Nc and f = f1Nc .

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

2. Show that fn f.

3. Show that

fnd

f d.


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

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

2. Show that gn lim inffn.3. Show that

gnd

fnd, for all n 1.

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

5. Show that

(lim inffn)d lim inf

fnd, and recall whyintegrals are well defined.

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

(lim infn+

fn)d lim infn+

fnd


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

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

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

3. Consider the equalities:

(f1A

)d = f dA = (f|A

)d|A

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

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


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5. Show that in order to prove (2), it is sufficient to consider tcase 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 msurable map, where (, F, ) is a measure space. let A F. We cpartial lebesgue integral of f with respect to over A, the integdenoted

A

f d, defined as:

A

f d= (f1A)d = f d

A = (f|A)d|Awhere A is the measure on (, F), A = (A ), f|A is the restrtion of f to A and |A is the restriction of to F|A, the trace ofon A.


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

A

ffor 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 msurable map, where (, F, ) is a measure space. Let : F [0, +

be defined by (A) =A f d, for all A F. Then, is a measure

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

gd =

gfd


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

L1R(, F, )=

f: (, F) (R, B(R)) measurable,

|f|d < +

L1C(, F, )

=

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()

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 combination

6. Show that L1C

(, F, ) is closed under C-linear combination


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Definition 47 Let u : R be a real-valued function defined oset . We call positive part and negative part of u the mapsand u respectively, defined as u+ = max(u, 0) and u = max(u,

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,

u

d,

v+

d,

v

dall well defined.

5. We define the integral off with respect to , denoted

f df d =

u+d

ud + i

v+d

vd

. Explain w

f d is a well defined complex number.


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

7. Show that

f d =

ud+i

vd and explain why all integrinvolved are well defined.

Definition 48 Let f = u + iv L1C

(, F, ) where (, F, ) imeasure space. We define the lebesgue integral of f with respec, denoted f d, as:

f d=

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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2. Show that f L1C

(, F, A).

3. Show that f|A L1C

(A, F|A, |A)

4. Show that (f1A)d = f dA = f|Ad|A.5. Show that 4. is:

A

u+d A

ud + iA

v+d A

vd

Definition 49 Let f L1C

(, F, ) , where (, F, ) is a measspace. let A F. We call partial lebesgue integral of f wrespect to over A, the integral denoted

A

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 restrtion of f to A and |A is the restriction of to F|A, the trace ofon A.


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Exercise 23. Let f, g L1R

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

1. Show that:

h+d + f

d + gd = h

d + f+d + g

+

2. Show that

hd =

f d +

gd.

3. Show that

(f)d =

f d

4. Show that if R then

(f)d =

f d.

5. Show that if f g then f d gd6. Show the following theorem.

Theorem 22 For all f, g L1C

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

f d +

gd


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Exercise 24. Let f, g be two maps, and (fn)n1 be a sequencemeasurable 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

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

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 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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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 quence of measurable maps fn : (, F) (C, B(C)) such that fn in C2 . Suppose that there exists some g L1

R(, F, ) such t

|fn| g for all n 1. Then f, fn L1C

(, F, ) for all n 1, an

limn+

|fn f|d = 0

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

f d. Let 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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3. Show that |

f d| =

(f)d.

4. Show that

(f)d =

ud.


Theorem 24 Let f L1C

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

f d

|f|d


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Tutorial 6: Product Spaces

6. Product Spaces

In the following, I is a non-empty set.

Definition 50 Let (i)iI be a family of sets, indexed by a n

empty set I. We call cartesian product of the family (i)iIset, denoted iIi, and defined by:

iI

i= { : I iIi , (i) i , i I}

In other words, iIi is the set of all maps defined on I, wvalues in iIi, such that (i) i for all i I.

Theorem 25 (Axiom of choice) Let (i)iI be a family of seindexed by a non-empty set I. Then, iIi is non-empty, if aonly if i is non-empty for all i I

1.

1When I is finite, this theorem is traditionally derived from other axioms.


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Exercise 1.

1. Let be a set and suppose that i = , i I. We use notation I instead of iIi. Show that

I is the set ofmaps : I .

2. What are the sets RR+

, RN , [0, 1]N , RR

?

3. Suppose I = N. We sometimes use the notation +n=1nstead of nNn. Let S be the set of all sequences (xn)nsuch that xn n for all n 1. Is S the same thing as tproduct +n=1n?

4. Suppose I = Nn = {1, . . . , n}, n 1. We use the notat1 . . . n instead of i{1,...,n}i. For 1 . . . nis customary to write (1, . . . , n) instead of , where we hai = (i). What is your guess for the definition of sets suchRn, R

n, Qn, Cn.

5. Let E , F , G be three sets. Define E F G.


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Definition 51 Let I be a non-empty set. We say that a familysets (I), where = , is a partition of I, if and only if:

(i) , I =

(ii) , , = I

I

=

(iii) I = I

Exercise 2. Let (i)iI be a family of sets indexed by I, and (I)be a partition of the set I.

1. For each , recall the definition of iIi.

2. Recall the definition of (iIi).

3. Define a natural bijection : iIi (iIi).

4. Define a naturalbijection : Rp Rn Rp+n, for all n, p


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Definition 52 Let (i)iI be a family of sets, indexed by a nempty set I. For all i I, let Ei be a set of subsets of i. We defia rectangle of the family (Ei)iI, as any subset A of iIi, of

form A = iIAi where Ai Ei {i} for all i I, and such t

Ai = i except for a finite number of indices i I. Consequently, set of all rectangles, denoted iIEi, is defined as:iI

Ei=

iI

Ai : Ai Ei {i} , Ai = i for finitely many i I

Exercise 3. (i)iI and (Ei)iI being as above:

1. Show that if I = Nn and i Ei for all i = 1, . . . , n, thE1 . . . En = {A1 . . . An : Ai Ei , i I}.

2. Let A be a rectangle. Show that there exists a finite subseof I such that: A = { iIi : (j) Aj , j J}some Aj s such that Aj Ej , for all j J.


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Definition 53 Let (i, Fi)iI be a family of measurable spaces, dexed by a non-empty set I. We call measurable rectangle , arectangle of the family (Fi)iI. The set of all measurable rectangis given by 2:

iI

Fi=iI

Ai : Ai Fi , Ai = i for finitely many i I

Definition 54 Let (i, Fi)iI be a family of measurable spaces, dexed by a non-empty set I. We define the product -algebra(Fi)iI, as the -algebra on iIi, denoted iIFi, and generaby all measurable rectangles, i.e.

iI

Fi=

iI

Fi

2Note that i Fi for all i I.


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Exercise 4.

1. Suppose I = Nn. Show that F1 . . . Fn is generated bysets of the form A1 . . .An, where Ai Fi for all i = 1, . . .

2. Show that B(R) B(R) B(R) is generated by sets of the foA B C where A ,B ,C B(R).

3. Show that if (, F) is a measurable space, B(R+) F is t-algebra on R+ generated by sets of the form B F whB B(R+) and F F.

Exercise 5. Let (i)iI be a family of non-empty sets and Ei b

subset of the power set P(i) for all i I.

1. Give a generator of the -algebra iI(Ei) on iIi.

2. Show that:

iI

Ei

iI

(Ei)


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3. Let A be a rectangle of the family ((Ei))iI. Show that ifAnot empty, then the representation A = iIAi with Ai (is unique. Define JA = {i I : Ai = i}. Explain why JA iwell-defined finite subset of I.

4. If A iI(Ei), Show that if A = , or A = and JA =then A (iIEi).

Exercise 6. Everything being as before, Let n 0. We assume ththe following induction hypothesis has been proved:

A iI(Ei), A = , cardJA = n A iIEiWe assume that A is a non empty measurable rectangle of ((Ei))with cardJA = n + 1. Let JA = {i1, . . . , in+1} be an extension ofFor all B i1 , we define:

AB=

iIAi


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where each Ai is equal to Ai except Ai1 = B. We define the set:

=

B i1 : A

B

iI

Ei

1. Show that Ai1 = , cardJAi1

= n and that Ai1 iI(E

2. Show that i1 .

3. Show that for all B i1 , we have Ai1\B = Ai1 \ AB .

4. Show that B i1 \ B .

5. Let Bn i1 , n 1. Show that ABn = n1ABn .

6. Show that is a -algebra on i1 .

7. Let B Ei1 , and for i I define Bi = i for all is excBi1 = B. Show that A

B = Ai1 (iIBi).

8. Show that (Ei1) .


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9. Show that A = AAi1 and A (iIEi).

10. Show that iI(Ei) (iIEi).

11. Show that (iIEi) = iI(Ei).

Theorem 26 Let (i)iI be a family of non-empty sets indexed bnon-empty set I. For all i I, let Ei be a set of subsets of i. Ththe product -algebra iI(Ei) on the cartesian product iIigenerated by the rectangles of (Ei)iI, i.e. :

iI

(Ei) = iI

EiExercise 7. Let TR denote the usual topology in R. Let n 1.

1. Show that TR . . . TR = {A1 . . . An : Ai TR}.

2. Show that B(R) . . . B(R) = (TR . . . TR).


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3. Define C2 = {]a1, b1] . . . ]an, bn] : ai, bi R}. Show tC2 S . . . S, where S = {]a, b] : a, b R}, but that tinclusion is strict.

4. Show that S . . . S (C2).

5. Show that B(R) . . . B(R) = (C2).

Exercise 8. Let and be two non-empty sets. Let A be a subof such that = A = . Let E = {A} P() and E = P(

1. Show that (E) = {, A , Ac, }.

2. Show that (E) = {, }.

3. Define C = {E F , E E, F E} and show that C = .

4. Show that E E = {A , }.

5. Show that (E) (E) = {, A , Ac , }.

6. Conclude that (E) (E) = (C) = {, }.


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Exercise 9. Let n 1 and p 1 be two positive integers.

1. Define F = B(R) . . . B(R)

n, and G = B(R) . . . B(R

pExplain why F G can be viewed as a -algebra on Rn+p.2. Show that F G is generated by sets of the form A1 . . .An

where Ai B(R), i = 1, . . . , n +p.

3. Show that:

B(R). . .B(R)

n+p= (B(R). . .B(R))

n(B(R). . .B(R

pExercise 10. Let (i, Fi)iI be a family of measurable spaces. L(I), where = , be a partition of I. Let = iIi a = (iIi).

1. Define a natural bijection between P() and P().


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2. Show that through such bijection, A = iIAi , whAi i, is identified with A = (iIAi)

.

3. Show that iIFi = (iIFi).

4. Show that iIFi = (iIFi).

Definition 55 Let be set and A be a set of subsets of . We ctopology generated by A, the topology on , denoted T(A), eqto the intersection of all topologies on , which contain A.

Exercise 11. Let be a set and A P().

1. Explain why T(A) is indeed a topology on .

2. Show that T(A) is the smallest topology T such that A T

3. Show that the metric topology on a metric space (E, d) is gerated by the open balls A = {B(x, ) : x E, > 0}.


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Definition 56 Let (i, Ti)iI be a family of topological spaces, dexed by a non-empty set I. We define the product topology(Ti)iI, as the topology on iIi, denoted iITi, and generatedall rectangles of (Ti)iI, i.e.

iI

Ti= T

iI

Ti

Exercise 12. Let (i, Ti)iI be a family of topological spaces.

1. Show that U iITi, if and only if:

x U , V iITi , x V U

2. Show that iITi iITi.

3. Show that iIB(i) = (iITi).

4. Show that iIB(i) B(iIi).


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Exercise 13. Let n 1 be a positive integer. For all x, y Rn,

(x, y)=

ni=1

xiyi

and we put x = (x, x).1. Show that for all t R, x + ty2 = x2 + t2y2 + 2t(x, y

2. From x + ty2 0 for all t, deduce that |(x, y)| x.y.

3. Conclude that x + y x + y.

Exercise 14. Let (1, T1), . . . , (n, Tn), n 1, be metrizable tological spaces. Let d1, . . . , dn be metrics on 1, . . . , n, inducing topologies T1, . . . , Tn respectively. Let = 1 . . . n and Tthe product topology on . For all x, y , we define:

d(x, y)=

ni=1

(di(xi, yi))2


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1. Show that d : R+ is a metric on .

2. Show that U is open in , if and only if, for all x U thare open sets U1, . . . , U n in 1, . . . , n respectively, such tha

x U1 . . . Un U

3. Let U T and x U. Show the existence of > 0 such tha

(i = 1, . . . , n di(xi, yi) < ) y U

4. Show that T Td .

5. let U Td

and x U. Show that existence of > 0 such thx B(x1, ) . . . B(xn, ) U

6. Show that Td T.

7. Show that the product topological space (, T) is metrizabl


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8. For all x, y , define:

d(x, y)=

ni=1

di(xi, yi)

d(x, y) = maxi=1,...,n

di(xi, yi)

Show that d, d are metrics on .

9. Show the existence of, , and > 0, such that we had d d and d d d.

10. Show that d

and d

also induce the product topology on

Exercise 15. Let (n, Tn)n1 be a sequence of metrizable topologispaces. For all n 1, let dn be a metric on n inducing the topoloTn. Let =

+n=1n be the cartesian product and T be the prod


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topology on . For all x, y , we define:

d(x, y)=

+n=1

1

2n(1 dn(xn, yn))

1. Show that for all a, b R+, we have 1 (a + b) 1 a + 1

2. Show that d is a metric on .

3. Show that U is open in , if and only if, for all x U, this an integer N 1 and open sets U1, . . . , U N in 1, . . . ,respectively, such that:

x U1 . . . UN +

n=N+1

n U

4. Show that d(x, y) < 1/2n dn(xn, yn) 2nd(x, y).

5. Show that for all U T and x U, there exists > 0 such thd(x, y) < y U.


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6. Show that T Td .

7. Let U Td and x U. Show the existence of > 0 and N such that:

Nn=1

12n

(1 dn(xn, yn)) < y U

8. Show that for all U Td and x U, there is > 0 and N such that:

x B(x1, ) . . . B(xN, ) +

n=N+1n U9. Show that Td T.

10. Show that the product topological space (, T) is metrizabl


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Definition 57 Let (, T) be a topological space. A subset H ofis called a countable base of (, T), if and only if H is at mcountable, and has the property:

U T , H H , U = VH

V

Exercise 16.

1. Show that H = {]r, q[ : r, q Q} is a countable base of (R, T

2. Show that if (, T) is a topological space with countable baand , then the induced topological space (, T|) a

has a countable base.3. Show that [1, 1] has a countable base.

4. Show that if (, T) and (S, TS) are homeomorphic, then (,has a countable base if and only if (S, TS) has a countable ba

5. Show that (R, TR) has a countable base.


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Exercise 17. Let (n, Tn)n1 be a sequence of topological spawith countable base. For n 1, Let {Vkn : k In} be a countabase of (n, Tn) where In is a finite or countable set. Let = n=1be the cartesian product and T be the product topology on .

all p 1, we define:

Hp=

Vk11 . . . V

kpp

+n=p+1

n : (k1, . . . , kp) I1 . . . Ip

and we put H = p1Hp.

1. Show that for all p 1, Hp T.

2. Show that H T.

3. For all p 1, show the existence of an injection jp : Hp N

4. Show the existence of a bijection 2 : N2 N.

5. For p 1, show the existence of an bijection p : Np N.


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6. Show that Hp is at most countable for all p 1.

7. Show the existence of an injection j : H N2.

8. Show that H is a finite or countable set of open sets in .

9. Let U T and x U. Show that there is p 1 and U1, . . . ,open sets in 1, . . . , p such that:

x U1 . . . Up +

n=p+1

n U

10. Show the existence of some Vx

H such that x Vx

U.

11. Show that H is a countable base of the topological space (, T

12. Show that +n=1B(n) B().

13. Show that H +n=1B(n).

14. Show that B() = +n=1B(n)


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Theorem 27 Let (n, Tn)n1 be a sequence of topological spawith countable base. Then, the product space (+n=1n,

+n=1Tn)

a countable base and:

B+n=1

n =+n=1

B(n)

Exercise 18.

1. Show that if (, T) has a countable base and n 1:

B(n) = B() . . . B() n

2. Show that B(Rn

) = B(R) . . . B(R).

3. Show that B(C) = B(R) B(R).


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Definition 58 We say that a metric space (E, d) is separableand only if there exists a finite or countable dense subset of E, a finite or countable subset A of E such that E = A, where A is closure of A in E.

Exercise 19. Let (E, d) be a metric space.

1. Suppose that (E, d) is separable. Let H = {B(xn,1

p) : n, p

where {xn : n 1} is a countable dense subset in E. Show thH is a countable base of the metric topological space (E, TdE

2. Suppose conversely that (E, T

d

E ) has a countable base H. Fall V H such that V = , take xV V. Show that the {xV : V H , V = } is at most countable and dense in E

3. For all x,y,x, y E, show that:

|d(x, y) d(x, y)| d(x, x) + d(y, y)


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4. Let TEE be the product topology on E E. Show that map d : (E E, TEE) (R+, TR+) is continuous.

5. Show that d : (E E, B(E E)) (R, B(R)) is measurabl

6. Show that d : (EE, B(E)B(E)) (R, B(R)) is measurabwhenever (E, d) is a separable metric space.

7. Let (, F) be a measurable space and f, g : (, F) (E, B(Ebe measurable maps. Show that : (, F) E E defined() = (f(), g()) is measurable with respect to the prod-algebra B(E) B(E).

8. Show that if (E, d) is separable, then : (, F) (R, B(Rdefined by () = d(f(), g()) is measurable.

9. Show that if (E, d) is separable then {f = g} F.

10. Let (En, dn)n1 be a sequence of separable metric spaces. Shthat the product space +n=1En is metrizable and separable


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Exercise 20. Prove the following theorem.

Theorem 28 Let (i, Fi)iI be a family of measurable spaces a(, F) be a measurable space. For all i I, let fi : i be a mand define f : iIi by f() = (fi())iI. Then, the map:

f : (, F)

iI

i,iI

Fi

is measurable, if and only if each fi : (, F) (i, Fi) is measura

Exercise 21.

1. Let , : R2

R with (x, y) = x + y and (x, y) = xShow that both and are continuous.

2. Show that , : (R2, B(R)B(R))(R, B(R)) are measurab

3. Let (, F) be a measurable space, and f, g : (, F) (R, B(Rbe measurable maps. Using the previous results, show that fand f.g are measurable with respect to F and B(R).


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Tutorial 7: Fubini Theorem

7. Fubini Theorem

Definition 59 Let (1, F1) and (2, F2) be two measurable spacLet E 1 2. For all 1 1, we call 1-section of E inthe set:

E1 = {2 2 : (1, 2) E}

Exercise 1. Let (1, F1) and (2, F2) be two measurable spacGiven 1 1, define:

1= {E 1 2 , E

1 F2}

1. Show that for all 1 1, 1 is a -algebra on 1 2.

2. Show that for all 1 1, F1 F2 1 .

3. Show that for all 1 1 and E F1 F2, we have E1

4. Show that the map 1E(1, ) is measurable with respto F2 and B(R), for all 1 1 and E F1 F2 .


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5. Let s be a simple function on (1 2, F1 F2). Show that all 1 1, the map s(1, ) is measurable with respto F2 and B(R).

6. Let f : (1 2, F1 F2) [0, +] be a non-negative, msurable map. Show that for all 1 1, the map f(1is measurable with respect to F2 and B(R).

7. Let f : (1 2, F1 F2) (R, B(R)) be a measurable mShow that for all 1 1, the map f(1, ) is measurawith respect to F2 and B(R).

8. Show the following theorem:

Theorem 29 Let (E, d) be a metric space, and (1, F1), (2, Fbe two measurable spaces. Let f : (1 2, F1 F2) (E, B(E))a measurable map . Then for all 1 1, the map f(1, )measurable with respect to F2 and B(E).


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Exercise 2. Let (i, Fi)iI be a family of measurable spaces wcardI 2. Let f : (iIi, iIFi) (E, B(E)) be a measuramap, where (E, d) be a metric space. Let i1 I. Put E1 = E1 = Fi1 , E2 = iI\{i1}i, E2 = iI\{i1}Fi.

1. Explain why f can be viewed as a map defined on E1 E2.

2. Show that f : (E1 E2, E1 E2) (E, B(E)) is measurable

3. For all i1 i1 , show that the map f(i1 , ) definediI\{i1}i is measurable w.r. to iI\{i1}Fi and B(E).

Definition 60 Let (, F, ) be a measure space. (, F, ) is saidbe a finite measure space, or we say that is a finite measuif and only if () < +.


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Definition 61 Let (, F, ) be a measure space. (, F, ) is sto be a -finite measure space, or a -finite measure, if aonly if there exists a sequence (n)n1 in F such that n a(n) < +, for all n 1.

Exercise 3. Let (, F, ) be a measure space.

1. Show that (, F, ) is -finite if and only if there exists a quence (n)n1 in F such that =

+n=1n, and (n) < +

for all n 1.

2. Show that if (, F, ) is finite, then has values in R+.

3. Show that if (, F, ) is finite, then it is -finite.

4. Let F : R R be a right-continuous, non-decreasing mShow that the measure space (R, B(R), dF) is -finite, whdF is the stieltjes measure associated with F.


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Exercise 4. Let (1, F1) be a measurable space, and (2, F2, 2)a -finite measure space. For all E F1 F2 and 1 1, define

E(1)=

2

1E(1, x)d2(x)

Let D be the set of subsets of 1 2, defined by:

D= {E F1 F2 : E : (1, F1) (R, B(R)) is measurable

1. Explain why for all E F1 F2, the map E is well define

2. Show that F1 F2 D.

3. Show that if2 is finite, A, B D and A B, then B \ A

4. Show that if En F1 F2, n 1 and En E, then En

5. Show that if2 is finite then D is a dynkin system on 1

6. Show that if2 is finite, then the map E : (1, F1) (R, B(Ris measurable, for all E F1 F2.


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7. Let (n2 )n1 in F2 be such that n2 2 and 2(

n2 ) < +

Define n2 = n22 = 2(

n2 ). For E F1 F2, we put:

nE(1)=

2

1E(1, x)dn2 (x)

Show that nE : (1, F1) (R, B(R)) is measurable, and:

nE(1) =

2

1n2 (x)1E(1, x)d2(x)

Deduce that nE E.

8. Show that the map E : (1, F1) (R, B(

R)) is measurabfor all E F1 F2.

9. Let s be a simple function on (1 2, F1 F2). Show tthe map

2

s(, x)d2(x) is well defined and measura

with respect to F1 and B(R).

10. Show the following theorem:


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Theorem 30 Let (1, F1) be a measurable space, and (2, F2,be a -finite measure space. Then for all non-negative and measura

map f : (1 2, F1 F2) [0, +], the map:

2 f(, x)d2(x)

is measurable with respect to F1 and B(R).

Exercise 5. Let (i, Fi)iI be a family of measurable spaces, wcardI 2. Let i0 I, and suppose that 0 is a -finite meason (i0 , Fi0). Show that if f : (iIi, iIFi) [0, +] is a n

negative and measurable map, then:

i0

f(, x)d0(x)

defined on iI\{i0}i, is measurable w.r. to iI\{i0}Fi and B(R


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Exercise 6. Let (1, F1, 1) and (2, F2, 2) be two -finite measspaces. For all E F1 F2, we define:

1 2(E)=

1

2

1E(x, y)d2(y)

d1(x)

1. Explain why 1 2 : F1 F2 [0, +] is well defined.

2. Show that 1 2 is a measure on F1 F2.

3. Show that if A B F1 F2, then:

1 2(A B) = 1(A)2(B)

Exercise 7. Further to ex. (6), suppose that : F1 F2 [0, +is another measure on F1 F2 with (A B) = 1(A)2(B), formeasurable rectangle A B. Let (n1 )n1 and (

n2 )n1 be sequen

in F1 and F2 respectively, such that n1 1,

n2 2, 1(

n1 ) < +

and 2(n2 ) < +. Define, for all n 1:

Dn= {E F1 F2 : (E (

n1

n2 )) = 1 2(E (

n1

n2


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1. Show that for all n 1, F1 F2 Dn.

2. Show that for all n 1, Dn is a dynkin system on 1 2.

3. Show that = 1 2.

4. Show that (12, F1F2, 12) is a -finite measure spa

5. Show that for all E F1 F2, we have:

1 2(E) =

2

1

1E(x, y)d1(x)

d2(y)

Exercise 8. Let (1, F1, 1), . . . , (n, Fn, n) be n -finite measspaces, n 2. Let i0 {1, . . . , n} and put E1 = i0 , E2 = i=i0E1 = Fi0 and E2 = i=i0Fi. Put 1 = i0 , and suppose that 2a -finite measure on (E2, E2) such that for all measurable rectani=i0Ai i=i0Fi, we have 2 (i=i0Ai) = i=i0i(Ai).


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1. Show that 1 2 is a -finite measure on the measure sp(1 . . . n, F1 . . . Fn) such that for all measurarectangles A1 . . . An, we have:

1 2(A1 . . . An) = 1(A1) . . . n(An)

2. Show by induction the existence of a measure on F1. . .Fsuch that for all measurable rectangles A1 . . . An, we ha

(A1 . . . An) = 1(A1) . . . n(An)

3. Show the uniqueness of such measure, denoted 1 . . . n

4. Show that 1 . . . n is -finite.

5. Let i0 {1, . . . , n}. Show that i0 (i=i0i) = 1 . . .


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Definition 62 Let (1, F1, 1), . . . , (n, Fn, n) be n -finite msure spaces, with n 2. We call product measure of 1, . . . ,the unique measure on F1 . . . Fn, denoted 1 . . . n, such tfor all measurable rectangles A1 . . . An in F1 . . . Fn, we ha

1 . . . n(A1 . . . An) = 1(A1) . . . n(An)

This measure is itself -finite.

Exercise 9. Prove that the following definition is legitimate:

Definition 63 We call lebesgue measure in Rn, n 1,

unique measure on (Rn

, B(Rn

)), denoted dx, dxn

or dx1 . . . d xn, suthat for all ai bi, i = 1, . . . , n, we have:

dx([a1, b1] . . . [an, bn]) =ni=1

(bi ai)


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Exercise 10.

1. Show that (Rn, B(Rn), dxn) is a -finite measure space.

2. For n, p 1, show that dxn+p = dxn dxp.

Exercise 11. Let (1, F1, 1) and (2, F2, 2) be -finite.

1. Let s be a simple function on (1 2, F1 F2). Show tha12

sd1 2 =

1

2

sd2

d1 =

2

1

sd1

d

2. Show the following:

Theorem 31 (Fubini) Let (1, F1, 1) and (2, F2, 2) be twofinite measure spaces. Let f : (1 2, F1 F2) [0, +] bnon-negative and measurable map. Then:

12

f d1 2 =

1

2

f d2

d1 =

2

1

f d1

d2


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Exercise 12. Let (1, F1, 1), . . . , (n, Fn, n) be n -finite measspaces, n 2. Let f : (1 . . . n, F1 . . . Fn) [0, +] bnon-negative, measurable map. Let be a permutation of Nn, i.ebijection from Nn to itself.

1. For all i=(1)i, define:

J1()=

(1)

f(, x)d(1)(x)

Explain why J1 : (i=(1)i, i=(1)Fi) [0, +] is a wdefined, non-negative and measurable map.

2. Suppose Jk : (i{(1),...,(k)}i, i{(1),...,(k)}Fi) [0, +is a

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