Functions between Measurable Spaces

Sometimes, it is  desirable to define measurability for a function f from one measurable space (X,X) into another measurable space (Y,Y). In this case one can define f to be measurable if and only if the set f^{-1} (E)=\{x\in X: f(x) \in E\} belongs to X for every set E belonging to Y.

This definition of measurability appears to differ from Definition 2.3 (earlier in the book), but Definition 2.3 is in fact equivalent to this definition in the case that Y=R and Y=B.

First, lets recap what is Definition 2.3:

A function f on X to R is said to be X-measurable (or simply measurable) if for every real number \alpha the set \{x\in X:f(x)>\alpha\} belongs to X.

Let (X,X) be a measurable space and f be a real-valued function defined on X. Then f is X-measurable if and only if f^{-1 }(E)\in X for every Borel set E.

Thus there is a close analogy between the measurable functions on a measurable space and continuous functions on a topological space.

Source:

The Elements of Integration and Lebesgue Measure

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