5. Lebesgue Integration - Probability

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).1. Show that is injective, and thats( ) is a finite subset ofR+.}

Tutorial 5: Lebesgue Integration 3 Definition 41 Let (Ω,F) be a measurable space, and s be a simple function on (Ω,F).Wecallpartition of the simple function s,any representation of the form: s = n i=1 αi1Ai where n ≥ 1, αi ∈ R+, Ai ∈Fand Ω=A1...An. Exercise 4. Lets bea simplefunction on (Ω,F) with twopartitions: s = n i=1 αi1Ai = m j=1 βj1Bj 1. Show that s = i,j αi1Ai∩Bj is a ...

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