Measurable function

Let $(X,M)$ and $(Y,N)$ be measure spaces. A map $f:X\to Y$ is $(M,N)$-measurable if $f^{-1}(E)\in M$ for all $E\in N.$

Examples of measurable functions

• A function $f:\mathbb {R} \to {\overline {\mathbb {R} }}$ is called a Lebesgue measurable function if $f$ is $(L,B_{\overline {\mathbb {R} }})$- measurable, where $L$ is the class of Lebesgue measurable sets and $B_{\overline {\mathbb {R} }}$ is Borel $\sigma$-algebra.
• A function $f:X\to Y$ is called Borel measurable if $f$ is $(B_{X},B_{Y})$-measurable.

Basic theorems of measurable functions

• Let $(X,M)$ and $(Y,N)$ be measure spaces. Suppose that $N$ is generated by a set $\varepsilon$. A map $f:X\to Y$ is $(M,N)$-measurable if $f^{-1}(E)\in M$ for all $E\in \varepsilon .$
• Let $(X,M)$, $(Y,N)$, and $(Z,P)$ be measure spaces. If a map $f:X\to Y$ is $(M,N)$-measurable and $g:Y\to Z$ is $(N,P)$-measurable, then $g\circ f:X\to Z$ is $(M,P)$-measurable.

If $N_{1}$ is another $\sigma$-algebra on $Y$ and a map $h:Y\to Z$ is $(N_{1},P)$-measurable, then $h\circ f:X\to Z$ is $(M,P)$-measurable when $N\subseteq N_{1}.$