---
tag: PHT84D91B7
kind: definition
name: Measurable function
msc: [28A20]
defines: YYNCJSR0PJ
uses: [N4KDT83V3S, 0WQXY32Q8D, 0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Measurable_function
---

## Definition

Let $(X, \Sigma)$ and $(Y, \mathcal{T})$ be [[measurable spaces|N4KDT83V3S]].
A [[function|0EPAT51DXQ]] $f \colon X \to Y$ is *measurable* if $f^{-1}(B) \in \Sigma$ for every $B \in \mathcal{T}$.
A function $f \colon X \to \R$ is measurable if it is measurable for the [[Borel sigma-algebra|0WQXY32Q8D]] on $\R$.

## Remarks

A function $f \colon X \to \R$ is measurable if and only if $\set{x \in X : f(x) > a} \in \Sigma$ for every $a \in \R$.
It suffices to check preimages of a family that generates $\mathcal{T}$.
Example: every continuous $f \colon \R \to \R$ is Borel measurable, since preimages of open sets are open and the open sets generate the Borel sets.
Example: the indicator function $\mathbf{1}_A$ is measurable if and only if $A \in \Sigma$.
Non-example: if $V$ is a non-measurable subset of $\R$, then $\mathbf{1}_V$ is not Lebesgue measurable, because $\set{\mathbf{1}_V > 1/2} = V$.
For Lebesgue measure on $\R$, a measurable function usually means one with $\Sigma$ the Lebesgue measurable sets and $\mathcal{T}$ the Borel sets.
A composition of measurable functions is measurable.