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 $E\in {\mathcal {N}}.$ This definition is analogous to the definition of a continuous function in a topological space, in which one requires that the preimage of each open set is open. In the case of the Lebesgue measure (or any Borel measure), all continuous functions are measurable.

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 the Borel $\sigma$-algebra on the extended real numbers.
• 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 ${\mathcal {E}}$. A map $f:X\to Y$ is $({\mathcal {M}},{\mathcal {N}})$-measurable if $f^{-1}(E)\in {\mathcal {M}}$ for all $E\in {\mathcal {E}}.$
• 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 ${\mathcal {N}}_{2}\subseteq {\mathcal {N}}_{1}.$

Properties of Borel Measurable functions

• If $(X,{\mathcal {M}})$ is a measure space and $f_{1},f_{2},...:X\to {\bar {\mathbb {R} }}$ measurable, then the following functions are measurable:
  • $f_{1}+f_{2}$
  • $f_{1}\cdot f_{2}$ where we define $0\cdot \pm \infty =0$
  • $f_{1}\vee f_{2}$ where $f_{1}\vee f_{2}(x)=\max\{f_{1}(x),f_{2}(x)\}$
  • $f_{1}\wedge f_{2}$ where $f_{1}\wedge f_{2}(x)=\min\{f_{1}(x),f_{2}(x)\}$
  • $\sup _{n}f_{n}=\vee _{n=1}^{\infty }f_{n}$
  • $\inf _{n}f_{n}=\wedge _{n=1}^{\infty }f_{n}$
  • $\limsup _{n}f_{n}$
  • $\liminf _{n}f_{n}$
  • $\lim _{n\to \infty }f_{n}$, if the limit exists for all $x\in X$
• If $f:X\to [0,\infty ]$ measurable, there exists a sequence $\{f_{n}\}$ of simple functions so that $f_{n}\nearrow f$ p.w., i.e. $f_{n}\leq f_{n+1}$

Non-measurable functions

It is possible to define functions that are not measurable. For example, if $A$ represents the Vitali set, then the indicator function $\mathbb {1} _{A}$ is not Borel-measurable. This follows directly from the fact that $\mathbb {1} _{A}^{-1}(\{1\})=A$, which is a non-measurable set, despite the fact that $\{1\}$ is closed in $\mathbb {R}$ (and hence $\{1\}\in {\mathcal {B}}_{\mathbb {R} }$).