Measurable functions

32 CHAPTER 3 Measurable functionsMeasurable functions in measure theory are analogous to continuous functionsin topology. A continuous function pulls back open sets to open sets, while ameasurable function pulls back Measurable sets to Measurable MeasurabilityMost of the theory of Measurable functions and integration does not dependon the specific features of the measure space on which the functions are defined, sowe consider general spaces, although one should keep in mind the case of functionsdefined onRorRnequipped with Lebesgue (X,A) and (Y,B) be Measurable spaces. A functionf:X Yis Measurable iff 1(B) Afor everyB that the measurability of a function depends only on the -algebras; it isnot necessary that any measures are order to show that a function is Measurable , it is sufficient to check themeasurability of the inverse images of sets that generate the -algebra on the that(X,A)and(Y,B)are Measurable spaces andB= (G)is generated by a familyG P(Y).

measurable if f 1(B) is a Lebesgue measurable subset of Rn for every Borel subset Bof R, and it is Borel measurable if f 1(B) is a Borel measurable subset of Rn for every Borel subset Bof R This de nition ensures that continuous functions f: Rn!R are Borel measur-able and functions that are equal a.e. to Borel measurable functions are Lebesgue ...

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  Functions, Measurable, Able, Measur, Measurable functions, Measur able

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