Measurable function

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

Examples of measurable functions

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

Basic theorems of measurable functions

• Let $(X,{\mathcal {M}})$ and $(Y,{\mathcal {N}})$ be measure spaces. Suppose that ${\mathcal {N}}$ is generated by a set $\varepsilon$. A map $f:X\to Y$ is $({\mathcal {M}},{\mathcal {N}})$-measurable if $f^{-1}(E)\in {\mathcal {M}}$ for all $E\in \varepsilon .$
• Let $(X,{\mathcal {M}})$, $(Y,{\mathcal {N}})$, and $(Z,{\mathcal {P}})$ be measure spaces. If a map $f:X\to Y$ is $({\mathcal {M}},{\mathcal {N}})$-measurable and $g:Y\to Z$ is $(N,{\mathcal {P}})$-measurable, then $g\circ f:X\to Z$ is $({\mathcal {M}},{\mathcal {P}})$-measurable. In particular, if $g:\mathbb {R} \to \mathbb {R}$ is Borel measurable and $f:\mathbb {R} \to \mathbb {R}$ is Lebesgue measurable, then $g\circ f$ is Lebesgue measurable.
• Let $(X,{\mathcal {M}})$, $(Y,{\mathcal {N}}_{1})$, $(Y,{\mathcal {N}}_{2})$, and $(Z,{\mathcal {P}})$ be measure spaces. If a map $f:X\to Y$ is $({\mathcal {M}},{\mathcal {N}}_{1})$-measurable and $g:Y\to Z$ is $({\mathcal {N}}_{2},{\mathcal {P}})$-measurable, then $g\circ f:X\to Z$ is $({\mathcal {M}},{\mathcal {P}})$-measurable when ${\mathcal {N}}_{2}\subseteq {\mathcal {N}}_{1}.$