Definition:Measurable Function/Real-Valued Function/Definition 1

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:

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