# Definition:Measurable Function

## Definition 1

Let $\struct {X, \Sigma}$ be a measurable space.

Let $E \in \Sigma$.

Then a function $f: E \to \R$ is said to be $\Sigma$-measurable on $E$ if and only if:

$\forall \alpha \in \R: \set {x \in E: \map f x \le \alpha} \in \Sigma$

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

## Definition 2

### Real-Valued Function

Let $\struct {X, \Sigma}$ be a measurable space.

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

A real-valued function $f: X \to \R$ is said to be ($\Sigma$-)measurable if and only if $f$ is $\Sigma \, / \, \BB$-measurable.

### Extended Real-Valued Function

Let $\struct {X, \Sigma}$ be a measurable space.

Let $\overline \BB$ 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 if and only if $f$ is $\Sigma \, / \, \overline \BB$-measurable.

### Positive Measurable Function

A measurable function $f$ is said to be a positive measurable function if and only if it also satisfies:

$f \ge 0$

where $\ge$ denotes pointwise inequality.