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 $E$ :


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

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

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 $f$ 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 $f$ is $\Sigma \, / \, \overline \BB$-measurable.

Positive Measurable Function
A measurable function $f$ is said to be a positive measurable function 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