Definition:Measurable Function/Real-Valued Function/Definition 3

Definition
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 $E$ one of the following holds:


 * $(1) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \le \alpha} \in \Sigma$
 * $(2) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x < \alpha} \in \Sigma$
 * $(3) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x \ge \alpha} \in \Sigma$
 * $(4) \quad$ $\forall \alpha \in \R : \set {x \in E : \map f x > \alpha} \in \Sigma$