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).}

2 Consider the map :s( ) {0,1}ndefined by (y)= ( y).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+.)

3 +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). 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).

4 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,+ ]. 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.

5 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. 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.

6 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). 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.

7 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. 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.

8 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.

9 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.

10 Exercise (fn)n 1be a sequence of non-negative and measur-able mapsfn:( ,F) [0,+ ]. 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).