5. Lebesgue Integration - Probability

1 Tutorial 5: Lebesgue Integration15. Lebesgue IntegrationIn the following, ( ,F, ) is a measure 39 LetA .Wecallcharacteristic functionofA,the map1A: R, defined by: ,1A( ) ={1if A0if AExercise , show that 1A:( ,F) ( R,B( R)) ismeasurable if and only ifA 40 Let( ,F)be a measurable space. We say that a maps: R+is asimple functionon( ,F),ifandonlyifsis ofthe form :s=n i=1 i1 Aiwheren 1, i R+andAi F, for alli=1,.., 5: Lebesgue Integration2 Exercise thats:( ,F) (R+,B(R+)) is measurable,wheneversis a simple function on ( ,F).Exercise a simple function on ( ,F) with representations= ni=1 i1Ai. Consider the map : {0,1}ndefined by ( )=(1A1( ),..,1An( )). For eachy s( ), pick one y such thaty=s( y). Consider the map :s( ) {0,1}ndefined by (y)= ( y).}

2 1. Show that is injective, and thats( ) is a finite subset ofR+.2. Show thats= s( ) 1{s= }3. Show that any simple functionscan be represented as:s=n i=1 i1 Aiwheren 1, i R+,Ai Fand =A1 .. 5: Lebesgue Integration3 Definition 41 Let( ,F)be a measurable space, andsbe a simplefunction on( ,F).Wecallpartitionof the simple functions,anyrepresentation of the form:s=n i=1 i1 Aiwheren 1, i R+,Ai Fand =A1 .. a simple function on ( ,F) with two partitions:s=n i=1 i1Ai=m j=1 j1Bj1. Show thats= i,j i1Ai Bjis a partition Recall the convention 0 (+ )=0and (+ )=+ if >0. For alla1,..,apin [0,+ ],p 1andx [0,+ ],prove the distributive property:x(a1+..+ap)=xa1+..+ 5: Lebesgue Integration43. Show that ni=1 i (Ai)= mj=1 j (Bj).4. Explain why the following definition is 42 Let( ,F, )be a measure space, andsbe a simplefunction on( ,F).

3 We define theintegralofswith respect to ,asthe sum, denotedI (s), defined by:I (s) =n i=1 i (Ai) [0,+ ]wheres= ni=1 i1 Aiis any partition 5: Lebesgue Integration5 Exercise , tbe two simple functions on ( ,F) with partitionss= ni=1 i1 Aiandt= mj=1 R+.1. Show thats+tis a simple function on ( ,F) with partition:s+t=n i=1m j=1( i+ j)1Ai Bj2. Show thatI (s+t)=I (s)+I (t).3. Show that sis a simple function on ( ,F).4. Show thatI ( s)= I (s).5. Why is the notationI ( s) meaningless if =+ or < Show that ifs tthenI (s) I (t). 5: Lebesgue Integration6 Exercise :( ,F) [0,+ ] be a non-negative and mea-surable map. For alln 1, we define:sn =n2n 1 k=0k2n1{k2n f<k+12n}+n1{n f}(1)1. Show thatsnis a simple function on ( ,F), for alln Show that equation (1) is a partitionsn, for alln Show thatsn sn+1 f, for alln Show thatsn fasn + for all , the sequence (sn( ))n 1is non-decreasing and convergestof( ) [0,+ ].

4 5: Lebesgue Integration7 Theorem 18 Letf:( ,F) [0,+ ]be a non-negative and mea-surable map, where( ,F)is a measurable space. There exists a se-quence(sn)n 1of simple functions on( ,F)such thatsn 43 Letf:( ,F) [0,+ ]be a non-negative andmeasurable map, where( ,F, )is a measure space. We define theLebesgue integraloffwith respect to ,denoted fd ,as: fd =sup{I (s):ssimple function on( ,F),s f}where, given any simple functionson( ,F),I (s)denotes its inte-gral with respect to .Exercise :( ,F) [0,+ ] be a non-negative and mea-surable Show that fd [0,+ ].2. Show that fd =I (f), wheneverfis a simple 5: Lebesgue Integration83. Show that gd fd , wheneverg:( ,F) [0,+ ]isnon-negative and measurable map withg Show that (cf)d =c fd ,if0<c<+ . Explain whyboth integrals are well defined.

5 Is the equality still true forc= Forn 1, putAn={f>1/n},andsn=(1/n)1An. Showthatsnis a simple function on ( ,F)withsn f. Show thatAn {f>0}.6. Show that fd =0 ({f>0})= Show that ifsis a simple function on ( ,F)withs f,then ({f>0}) = 0 impliesI (s)= Show that fd =0 ({f>0})= Show that (+ )fd =(+ ) fd . Explain why both inte-grals are well 5: Lebesgue Integration910. Show that (+ )1{f=+ } fand: (+ )1{f=+ }d =(+ ) ({f=+ })11. Show that fd <+ ({f=+ })= Suppose that ( ) = + and takef= 1. Show that theconverse of the previous implication is not a simple function on ( ,F). LetA Show thats1 Ais a simple function on ( ,F).2. Show that for any partitions= ni=1 i1 Aiofs,wehave:I (s1A)=n i=1 i (Ai A) 5: Lebesgue Integration103. Let :F [0,+ ] be defined by (A)=I (s1A).

6 Show that is a measure SupposeAn F,An A. Show thatI (s1An) I (s1A).Exercise (fn)n 1be a sequence of non-negative and measur-able mapsfn:( ,F) [0,+ ], such thatfn Recall what the notationfn Explain whyf:( ,F) ( R,B( R)) is Let =supn 1 fnd . Show that fnd .4. Show that fd .5. Letsbe any simple function on ( ,F) such thats ]0,1[. Forn 1, defineAn={cs fn}. Show thatAn FandAn . 5: Lebesgue Integration116. Show thatcI (s1An) fnd , for alln Show thatcI (s) .8. Show thatI (s) .9. Show that fd .10. Conclude that fnd fd .Theorem 19 (Monotone Convergence)Let( ,F, )be a mea-sure space. Let(fn)n 1be a sequence of non-negative and measurablemapsfn:( ,F) [0,+ ]such thatfn fnd fd .Exercise , g:( ,F) [0,+ ] be two non-negative andmeasurable maps.

7 Leta, b [0,+ ]. 5: Lebesgue Integration121. Show that if (fn)n 1and (gn)n 1are two sequences of non-negative and measurable maps such thatfn fandgn g,thenfn+gn f+ Show that (f+g)d = fd + gd .3. Show that (af+bg)d =a fd +b gd .Exercise (fn)n 1be a sequence of non-negative and mea-surable mapsfn:( ,F) [0,+ ]. Definef= + n= Explain whyf:( ,F) [0,+ ] is well defined, non-negativeand Show that fd = + n=1 fnd . 5: Lebesgue Integration13 Definition 44 Let( ,F, )be a measure space and letP( )be aproperty depending on . We say that the propertyP( )holds -almost surely, and we writeP( ) , if and only if: N F, (N)=0, Nc,P( )holdsExercise ( ) be a property depending on , such that{ :P( )holds}is an element of the Show thatP( ), ({ :P( )holds}c)= Explain why in general, the right-hand side of this equivalencecannot be used to defined -almost sure ( ,F, ) be a measure space and (An)n 1be asequence of elements ofF.

8 Show that ( + n=1An) + n=1 (An). 5: Lebesgue Integration14 Exercise (fn)n 1be a sequence of mapsfn: [0,+ ].1. Translate formally the statementfn f Translate formallyfn f and n,(fn fn+1 )3. Show that the statements thatf, g:( ,F) [0,+ ] are non-negativeand measurable withf=g LetN F, (N) = 0 such thatf=gonNc. Explain why fd = (f1N)d + (f1Nc)d ,allintegrals being well defined. Show that fd = gd .Exercise (fn)n 1is a sequence of non-negative andmeasurable maps andfis a non-negative and measurable map, suchthatfn f LetN F, (N) = 0, such thatfn fn=fn1 Ncand f= Explain why fand the fn s are non-negative and 5: Lebesgue Integration152. Show that fn Show that fnd fd .Exercise (fn)n 1be a sequence of non-negative and measur-able mapsfn:( ,F) [0,+ ].

9 Forn 1, we definegn=infk Explain why thegn s are non-negative and Show thatgn lim Show that gnd fnd , for alln Show that if (un)n 1and (vn)n 1are two sequences in Rwithun vnfor alln 1, then lim infun lim Show that (lim inffn)d lim inf fnd , and recall why allintegrals are well 5: Lebesgue Integration16 Theorem 20 (Fatou Lemma)Let( ,F, )be a measure space,and let(fn)n 1be a sequence of non-negative and measurable mapsfn:( ,F) [0,+ ].Then: (lim infn + fn)d lim infn + fnd Exercise :( ,F) [0,+ ] be a non-negative and mea-surable map. LetA Recall what is meant by the induced measure space (A,F|A, |A).Why is it important to haveA F. Show that the restrictionofftoA,f|A:(A,F|A) [0,+ ] is We define the map A:F [0,+ ]by A(E)= (A E), forallE F.

10 Show that ( ,F, A) is a measure Consider the equalities: (f1A)d = fd A= (f|A)d |A(2) 5: Lebesgue Integration17 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 Show that in order to prove (2), it is sufficient to consider thecase whenfis a simple function on ( ,F).5. Show that in order to prove (2), it is sufficient to consider thecase whenfis of the formf=1B,forsomeB Show that (2) is indeed 5: Lebesgue Integration18 Definition 45 Letf:( ,F) [0,+ ]be a non-negative and mea-surable map, where( ,F, )is a measure space. letA Lebesgue integraloffwith respect to overA, the integraldenoted Afd , defined as: Afd = (f1A)d = fd A= (f|A)d |Awhere Ais the measure on( ,F), A= (A ),f|Ais the restric-tion offtoAand |Ais the restriction of toF|A, the trace , g:( ,F) [0,+ ] be two non-negative andmeasurable maps.