# Definition:Measurable Function

## Contents

## Definition 1

Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $E \in \Sigma$.

Then a function $f: E \to \R$ is said to be **$\Sigma$-measurable on $E$** iff:

- $\forall \alpha \in \R: \left\{{x \in E : f \left({x}\right) \le \alpha}\right\} \in \Sigma$

See the theorem on measurable images for equivalences of this definition.

## Definition 2

### Real-Valued Function

Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $\mathcal B$ be the Borel $\sigma$-algebra on $\R$.

A real-valued function $f: X \to \R$ is said to be **($\Sigma$-)measurable** iff $f$ is $\Sigma \, / \, \mathcal B$-measurable.

### Extended Real-Valued Function

Let $\left({X, \Sigma}\right)$ be a measurable space.

Let $\overline{\mathcal B}$ be the Borel $\sigma$-algebra on the extended real number space.

An extended real-valued function $f: X \to \R$ is said to be **($\Sigma$-)measurable** iff $f$ is $\Sigma \, / \, \overline{\mathcal B}$-measurable.

### Positive Measurable Function

A measurable function $f$ is said to be a **positive measurable function** iff it also satisfies:

- $f \ge 0$

where $\ge$ denotes pointwise inequality.

## Also see

- Measurable Mapping, a more general notion of which this is a specialization.
- Space of Measurable Functions, naming collections of
**$\Sigma$-measurable functions**conveniently

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $\S 8$