Definition:Measurable Function

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.