Measurable Maps

Definition (Measurable Map)

Suppose $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces. We say that a map $f:X\to Y$ is measurable (with respect to $\mathcal{A}$ and $\mathcal{B}$) if $f^{-1}(B)\in \mathcal{A}$ for all $B\in \mathcal{B}$.

In other words, a map between measurable spaces is said to be measurable iff preimages of measurable sets are measurable.

The composition of measurable maps is measurable.

It is sufficient to check measurability for a generator:

Suppose that $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are measurable spaces, and that $\mathcal{E}$ is a generator of $\mathcal{B}$. Then a map $f:X\to Y$ is measurable iff $f^{-1}(E)\in \mathcal{A}$ for every $E\in \mathcal{E}$.