6. Product Spaces - Probability

1 Tutorial 6: Product the following,Iis a non-empty 50 Let( i)i Ibe a family of sets, indexed by a non-empty productof the family( i)i Itheset, denoted i I i, and defined by: i I i ={ :I i I i, (i) i, i I}In other words, i I iis the set of all maps defined onI,withvalues in i I i, such that (i) ifor alli (Axiomofchoice)Let( i)i Ibe a family of sets,indexed by a non-empty , i I iis non-empty, if andonly if iis non-empty for alli finite, this theorem is traditionally derived from other 6: Product Spaces2 Exercise Let be a set and suppose that i= , i Iinstead of i I i. Show that Iis the set of allmaps :I .2.

2 What are the setsRR+,RN,[0,1]N, RR?3. SupposeI=N . We sometimes use the notation + n=1 nin-stead of n N the set of all sequences (xn)n 1such thatxn nfor alln 1. IsSthesamethingastheproduct + n=1 n?4. SupposeI=Nn={1,..,n},n 1. We use the notation 1 .. ninstead of i {1,..,n} 1 .. n,itis customary to write ( 1,.., n) instead of , where we have i= (i). What is your guess for the definition of sets such asRn, Rn,Qn, LetE, F, Gbe three sets. DefineE F 6: Product Spaces3 Definition 51 LetIbe a non-empty set. We say that a family ofsets(I ) ,where = ,isapartitionofI,ifandonlyif:(i) ,I = (ii) , , = I I = (iii)I= I Exercise ( i)i Ibe a family of sets indexed byI,and(I ) be a partition of the For each , recall the definition of i I Recall the definition of ( i I i).

3 3. Define anaturalbijection : i I i ( i I i).4. Define anaturalbijection :Rp Rn Rp+n, for alln, p 6: Product Spaces4 Definition 52 Let( i)i Ibe a family of sets, indexed by a non-empty setI. For alli I,letEibe a set of subsets of i. We definearectangleof the family(Ei)i I, as any subsetAof i I i,oftheformA= i IAiwhereAi Ei { i}for alli I, and such thatAi= iexcept for a finite number of indicesi I. Consequently, theset of all rectangles, denoted i IEi, is defined as: i IEi ={ i IAi:Ai Ei { i},Ai = ifor finitely manyi I}Exercise 3.( i)i Iand (Ei)i Ibeing as above:1. Show that ifI=Nnand i Eifor alli=1,..,n,thenE1 .. En={A1.}

4 An:Ai Ei, i I}.2. LetAbe a rectangle. Show that there exists a finite subsetJofIsuch that:A={ i I i: (j) Aj, j J}forsomeAj s such thatAj Ej, for allj 6: Product Spaces5 Definition 53 Let( i,Fi)i Ibe a family of measurable Spaces , in-dexed by a non-empty rectangle,anyrectangle of the family(Fi)i I. The set of all measurable rectanglesis given by2: i IFi ={ i IAi:Ai Fi,Ai = ifor finitely manyi I}Definition 54 Let( i,Fi)i Ibe a family of measurable Spaces , in-dexed by a non-empty setI. We define theproduct -algebraof(Fi)i I,asthe -algebra on i I i,denoted i IFi, and generatedby all measurable rectangles, i IFi = ( i IFi)2 Note that i Fifor alli 6: Product Spaces6 Exercise SupposeI=Nn.

5 Show thatF1 .. Fnis generated by allsets of the formA1 .. An,whereAi Fifor alli=1,.., Show thatB(R) B(R) B(R) is generated by sets of the formA B CwhereA, B, C B(R).3. Show that if ( ,F) is a measurable space ,B(R+) Fis the -algebra onR+ generated by sets of the formB FwhereB B(R+)andF ( i)i Ibe a family of non-empty sets andEibe asubset of the power setP( i) for alli Give a generator of the -algebra i I (Ei)on i I Show that: ( i IEi) i I (Ei) 6: Product Spaces73. LetAbe a rectangle of the family ( (Ei))i I. Show that ifAisnot empty, then the representationA= i IAiwithAi (Ei)is unique. DefineJA={i I:Ai = i}. Explain whyJAis awell-defined finite subset IfA i I (Ei), Show that ifA= ,orA = andJA= ,thenA ( i IEi).

6 Exercise being as before, Letn 0. We assume thatthe following induction hypothesis has been proved:A i I (Ei),A = ,cardJA=n A ( i IEi)We assume thatAis a non empty measurable rectangle of ( (Ei))i Iwith cardJA=n+1. LetJA={i1,..,in+1}be an extension 6: Product Spaces8 For allB i1, we define:AB = i I Aiwhere each Aiis equal toAiexcept Ai1=B. Wedefinetheset: ={B i1:AB ( i IEi)}1. Show thatA i1 = ,cardJA i1=nand thatA i1 i I (Ei).2. Show that i1 .3. Show that for allB i1,wehaveA i1\B=A i1\ Show thatB i1\B .5. LetBn i1,n 1. Show thatA Bn= n Show that is a -algebra on 6: Product Spaces97. LetB Ei1,andfori Idefine Bi= ifor alli s except Bi1=B.

7 Show thatAB=A i1 ( i I Bi).8. Show that (Ei1) .9. Show thatA=AAi1andA ( i IEi).10. Show that i I (Ei) ( i IEi).11. Show that ( i IEi)= i I (Ei).Theorem 26 Let( i)i Ibe a family of non-empty sets indexed by anon-empty setI. For alli I,letEibe a set of subsets of ,the Product -algebra i I (Ei)on the Cartesian Product i I iisgenerated by the rectangles of(Ei)i I, : i I (Ei)= ( i IEi) 6: Product Spaces10 Exercise the usual topology Show thatTR .. TR={A1 .. An:Ai TR}.2. Show thatB(R) .. B(R)= (TR .. TR).3. DefineC2={]a1,b1] .. ]an,bn]:ai,bi R}. Show thatC2 S .. S,whereS={]a, b]:a, b R}, but that theinclusion is Show thatS .. S (C2).

8 5. Show thatB(R) .. B(R)= (C2).Exercise and be two non-empty sets. LetAbe a subsetof such that =A = . LetE={A} P( ) andE = P( ).1. Show that (E)={ ,A,Ac, }.2. Show that (E )={ , }. 6: Product Spaces113. DefineC={E F,E E,F E }and show thatC= .4. Show thatE E ={A , }.5. Show that (E) (E )={ ,A ,Ac , }.6. Conclude that (E) (E ) = (C)={ , }.Exercise 1andp 1 be two positive DefineF=B(R) .. B(R) n,andG=B(R) .. B(R) whyF Gcanbeviewedasa -algebra onRn+ Show thatF Gis generated by sets of the formA1 .. An+pwhereAi B(R),i=1,..,n+ 6: Product Spaces123. Show that:B(R) .. B(R) n+p=(B(R) .. B(R)) n (B(R).)

9 B(R)) pExercise ( i,Fi)i Ibe a family of measurable Spaces . Let(I ) ,where = , be a partition = i I iand = ( i I i).1. Define anaturalbijection betweenP( ) andP( ).2. Show that through such bijection,A= i IAi , whereAi i,isidentifiedwithA = ( i I Ai) .3. Show that i IFi= ( i I Fi).4. Show that i IFi= ( i I Fi). 6: Product Spaces13 Definition 55 Let be set andAbe a set of subsets of .Wecalltopology generatedbyA, the topology on ,denotedT(A),equalto the intersection of all topologies on , which be a set andA P( ).1. Explain whyT(A) is indeed a topology on .2. Show thatT(A) is the smallest topologyTsuch thatA Show that the metric topology on a metric space (E, d)isgen-erated by the open ballsA={B(x, ):x E, >0}.

10 Definition 56 Let( i,Ti)i Ibe a family of topological Spaces , in-dexed by a non-empty setI. We define theproduct topologyof(Ti)i I, as the topology on i I i,denoted i ITi, and generated 6: Product Spaces14all rectangles of(Ti)i I, i ITi =T( i ITi)Exercise ( i,Ti)i Ibe a family of topological Show thatU i ITi, if and only if: x U, V i ITi,x V U2. Show that i ITi i Show that i IB( i)= ( i ITi).4. Show that i IB( i) B( i I i). 6: Product Spaces15 Exercise 1 be a positive integer. For allx, y Rn,let:(x, y) =n i=1xiyiand we put x = (x, x).1. Show that for allt R, x+ty 2= x 2+t2 y 2+2t(x, y).2. From x+ty 2 0 for allt, deduce that|(x, y)| x.