2. Caratheodory’s Extension - Probability Tutorials

1 tutorial 2 : caratheodory s Extension12. caratheodory s ExtensionIn the following, is a set. Whenever a union of sets is denoted asopposed to , it indicates that the sets involved are pairwise 6 Asemi-ringon is a subsetSof the power setP( )with the following properties:(i) S(ii)A, B S A B S(iii)A, B S n 0, Ai S:A\B=n i=1 AiThe last property (iii) says that wheneverA, B S,thereisn 0andA1,..,AninSwhich are pairwise disjoint, such thatA\B=A1 .. 0, it is understood that the corresponding unionis equal to , (in which caseA B). 2: caratheodory s Extension2 Definition 7 Aringon is a subsetRof the power setP( )withthe following properties:(i) R(ii)A, B R A B R(iii)A, B R A\B RExercise thatA B=A\(A\B) and therefore that aring is closed under pairwise that a ring on is also a semi-ring on .Exercise that a set can be decomposed as =A1 A2 A3whereA1,A2andA3are distinct from and . DefineS1 ={ ,A1,A2,A3, }andS2 ={ ,A1,A2 A3, }. Show thatS1andS2are semi-rings on , but thatS1 S2fails to be a semi-ringon.

2 Exercise (Ri)i Ibe an arbitrary family of rings on , withI = . Show thatR = i IRiisalsoaringon . 2: caratheodory s Extension3 Exercise a subset of the power setP( ). Define:R(A) ={Rring on :A R}Show thatP( ) is a ring on , and thatR(A) is not empty. Define:R(A) = R R(A)RShow thatR(A) is a ring on such thatA R(A),andthatitisthe smallest ring on with such property, ( ifRis a ring on andA RthenR(A) R).Definition 8 LetA P( ).Wecallring generatedbyA,thering on ,denotedR(A), equal to the intersection of all rings on ,which a semi-ring on . Define the setRof all finiteunions of pairwise disjoint elements ofS, ={A:A= ni=1 Aifor somen 0,Ai S} 2: caratheodory s Extension4(where ifn= 0, the corresponding union is empty, R). LetA= ni=1 AiandB= pj=1Bj R:1. Show thatA B= i,j(Ai Bj)andthatRis closed underpairwise Show that ifp 1thenA\B= pj=1( ni=1(Ai\Bj)).3. Show thatRis closed under pairwise Show thatA B=(A\B) Band conclude thatRis a ringon .5. Show thatR(S)= being as before, define:R ={A:A= ni=1 Aifor somen 0,Ai S} 2: caratheodory s Extension5(We do not require the sets involved in the union to be pairwise dis-joint).

3 Using the fact thatRis closed under finite union, show thatR R, and conclude thatR =R=R(S).Definition 9 LetA P( )with ,any map :A [0,+ ]with the following properties:(i) ( )=0(ii)A A,An AandA=+ n=1An (A)=+ n=1 (An)The indicates that we assume theAn s to be pairwise disjoint inthe of (ii). It is customary to say in view of condition (ii)thatameasureiscountably a -algebra on explain why property (ii)canbe replaced by:(ii) An AandA=+ n=1An (A)=+ n=1 (An) 2: caratheodory s Extension6 Exercise P( ) with Aand :A [0,+ ]beameasure Show that ifA1,..,An Aare pairwise disjoint and the unionA= ni=1 Ailies inA,then (A)= (A1)+..+ (An).2. Show that ifA, B A,A BandB\A Athen (A) (B).Exercise a semi-ring on , and :S [0,+ ]beameasure onS. Suppose that there exists an Extension of onR(S), a measure :R(S) [0,+ ] such that |S= .1. LetAbe an element ofR(S) with representationA= ni=1 Aias a finite union of pairwise disjoint elements ofS. Show that (A)= ni=1 (Ai)2.

4 Show that if :R(S) [0,+ ] is another measure with |S= , another Extension of onR(S), then = . 2: caratheodory s Extension7 Exercise a semi-ring on and :S [0,+ ]beameasure. LetAbe an element ofR(S) with two representations:A=n i=1Ai=p j=1 Bjas a finite union of pairwise disjoint elements Fori=1,..,n, show that (Ai)= pj=1 (Ai Bj)2. Show that ni=1 (Ai)= pj=1 (Bj)3. Explain why we can define a map :R(S) [0,+ ]as: (A) =n i=1 (Ai)4. Show that ( )= 2: caratheodory s Extension8 Exercise being as before, suppose that (An)n 1isa sequence of pairwise disjoint elements ofR(S), eachAnhaving therepresentation:An=pn k=1 Akn,n 1as a finite union of disjoint elements ofS. Suppose moreover thatA= + n=1 Anis an element ofR(S) with representationA= pj=1Bj,as a finite union of pairwise disjoint elements Show that forj=1,..,p,Bj= + n=1 pnk=1(Akn Bj)andexplain whyBjis of the formBj= + m=1 Cmfor some sequence(Cm)m 1of pairwise disjoint elements Show that (Bj)= + n=1 pnk=1 (Akn Bj)3.

5 Show that forn 1andk=1,..,pn,Akn= pj=1(Akn Bj) 2: caratheodory s Extension94. Show that (Akn)= pj=1 (Akn Bj)5. Recall the definition of of exercise (11) and show that it is ameasure onR(S).Exercise the following theorem:Theorem 2 LetSbe a semi-ring on .Let :S [0,+ ]be ameasure onS. There exists a unique measure :R(S) [0,+ ]such that |S= . 2: caratheodory s Extension10 Definition 10We define anouter-measureon as being anymap :P( ) [0,+ ]with the following properties:(i) ( )=0(ii)A B (A) (B)(iii) (+ n=1An) + n=1 (An)Exercise that (A B) (A)+ (B), where isan outer-measure on andA, B .Definition 11 Let be an outer-measure on . We define: ( ) ={A : (T)= (T A)+ (T Ac), T }We call ( )the -algebra associatedwith the outer-measure .Note that the fact that ( ) is indeed a -algebra on , remains tobe proved. This will be your task in the following 2: caratheodory s Extension11 Exercise be an outer-measure on . Let = ( )bethe -algebra associated with.

6 LetA, B andT 1. Show that andAc .2. Show that (T A)= (T A B)+ (T A Bc)3. Show thatT Ac=T (A B)c Ac4. Show thatT A Bc=T (A B)c A5. Show that (T Ac)+ (T A Bc)= (T (A B)c)6. Adding (T (A B)) on both sides 5., conclude thatA B .7. Show thatA BandA\Bbelong to .Exercise being as before, letAn ,n 1. DefineB1=A1andBn+1=An+1\(A1 .. An). Show that theBn s arepairwise disjoint elements of and that + n=1An= + n= 2: caratheodory s Extension12 Exercise being as before, show that ifB, C andB C= ,then (T (B C)) = (T B)+ (T C) for anyT .Exercise being as before, let (Bn)n 1be a sequenceof pairwise disjoint elements of , and letB = + n= Explain why Nn=1Bn 2. Show that (T ( Nn=1Bn)) = Nn=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) (T Bc)+ (T B) (T Bc)+ + n=1 (T Bn)6. Show thatB and (B)= + n=1 (Bn).7. Show that is a -algebra on , and | is a measure on . 2: caratheodory s Extension13 Theorem 3 Let :P( ) [0,+ ]be an outer-measure on.

7 Then ( ), the so-called -algebra associatedwith , is indeed a -algebra on and | ( ),isameasureon ( ).Exercise a ring on and :R [0,+ ]beameasure onR. For allT , define: (T) =inf{+ n=1 (An),(An)isanR-cover ofT}where anR-cover ofTis defined as any sequence (An)n 1of elementsofRsuch thatT + n= =+ .1. Show that ( )= Show that ifA Bthen (A) (B). 2: caratheodory s Extension143. Let (An)n 1be a sequence of subsets of , with (An)<+ for alln 1. Given >0, show that for alln 1, there existsanR-cover (Apn)p 1ofAnsuch that:+ p=1 (Apn)< (An)+ /2nWhy is it important to assume (An)<+ .4. Show that there exists anR-cover (Rk)of + n=1 Ansuch that:+ k=1 (Rk)=+ n=1+ p=1 (Apn)5. Show that ( + n=1An) + + n=1 (An)6. Show that is an outer-measure on . 2: caratheodory s Extension15 Exercise being as before, LetA (An)n 1be anR-cover ofAand putB1=A1 A, and:Bn+1 =(An+1 A)\((A1 A) .. (An A))1. Show that (A) (A).2. Show that (Bn)n 1is a sequence of pairwise disjoint elementsofRsuch thatA= + n= Show that (A) (A) and conclude that |R=.

8 Exercise being as before, LetA RandT .1. Show that (T) (T A)+ (T Ac).2. Let (Tn)beanR-cover ofT. Show that (Tn A)and(Tn Ac)areR-covers ofT AandT Show that (T A)+ (T Ac) (T). 2: caratheodory s Extension164. Show thatR ( ).5. Conclude that (R) ( ).Exercise the following theorem:Theorem 4 ( caratheodory s Extension )LetRbe a ring on and :R [0,+ ]be a measure onR. There exists a measure : (R) [0,+ ]such that |R= .Exercise a semi-ring on . Show that (R(S)) = (S).Exercise the following theorem:Theorem 5 LetSbe a semi-ring on and :S [0,+ ]be ameasure onS. There exists a measure : (S) [0,+ ]such that |S=.