# Measurable Maps

Definition (Measurable Map)

Suppose $(X,A)$ and $(Y,B)$ are measurable spaces. We say that a map $f:X→Y$ is

measurable(with respect to $A$ and $B$) if $f_{−1}(B)∈A$ for all $B∈B$.

The composition of measurable maps is measurable.

It is sufficient to check measurability for a generator:

Suppose that $(X,A)$ and $(Y,B)$ are measurable spaces, and that $E$ is a generator of $B$. Then a map $f:X→Y$ is measurable iff $f_{−1}(E)∈A$ for every $E∈E$.