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

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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 $E$ if and only if one of the following holds:

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