\begin{definition}[Definition:Measurable Function/Real-Valued 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 $E$''' {{iff}}:
:$\forall \alpha \in \R: \set {x \in E: \map f x \le \alpha} \in \Sigma$
Category:Definitions/Measurable Functions
\end{definition}