Chapter 4 Measurable Functions - LSU Math

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Chapter 4Measurable FunctionsIfXis a set andA P(X) is a -field, then (X,A) is called ameasurablespace. If is a countably additive measure defined onAthen (X,A, )is called ameasure space. In this Chapter we will introduce the family ofmeasurable functionsfor which we will seek to define the Lebesgue will prove the very important fact that pointwise limits of measurablefunctions must be Measurable . This is encouraging because pointwise limitsof Riemann integrable Functions need not be Riemann Measurable FunctionsDefinition (X,A, ) be a measure Iff:X Rwe say thatfisA-measurableprovided thatf 1( ,a) ={x X|f(x)< a} Afor alla Iff:X C, the complex numbers, we writef(x) =u(x) +iv(x) forreal-valued functionsu= fandv= f. We sayfisA-measurableprovided that Iff:X S, whereSis atopological space, we say thatfisA- Measurable provided thatf 1(G) Afor everyopensetG Example definition is motivated by Section is necessary to prove that the three parts of the definitionof measura-bility of a function are that iff:X RisA- Measurable , thenf 1(G) Afor every open setG R.

Chapter 4 Measurable Functions If Xis a set and A ⊆ P(X) is a σ-field, then (X,A) is called a measurable space. If µis a countably additive measure defined on A then (X,A,µ) is called a measure space. In this chapter we will introduce the family of

  Chapter, Functions