\begin{definition}[Definition:Measurable Function/Real-Valued Function/Definition 3]
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}} one of the following holds:
:$(1) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \le \alpha} \in \Sigma$
:$(2) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x < \alpha} \in \Sigma$
:$(3) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \ge \alpha} \in \Sigma$
:$(4) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x > \alpha} \in \Sigma$
Category:Definitions/Measurable Functions
\end{definition}