# Characterization of Measurable Functions

## Theorem

Let $\struct {X, \Sigma}$ be a measurable space.

Let $f: X \to \overline \R$ be an extended real-valued function.

Then the following are all equivalent:

 $\text {(1)}: \quad$ $\ds$  $\ds f$ is measurable  $\text {(2)}: \quad$ $\ds$  $\ds \forall \alpha \in \R: \set {x \in X: \map f x \le \alpha} \in \Sigma$ $\text {(2')}: \quad$ $\ds$  $\ds \forall \alpha \in \Q: \set {x \in X: \map f x \le \alpha} \in \Sigma$ $\text {(3)}: \quad$ $\ds$  $\ds \forall \alpha \in \R: \set {x \in X: \map f x < \alpha} \in \Sigma$ $\text {(3')}: \quad$ $\ds$  $\ds \forall \alpha \in \Q: \set {x \in X: \map f x < \alpha} \in \Sigma$ $\text {(4)}: \quad$ $\ds$  $\ds \forall \alpha \in \R: \set {x \in X: \map f x \ge \alpha} \in \Sigma$ $\text {(4')}: \quad$ $\ds$  $\ds \forall \alpha \in \Q: \set {x \in X: \map f x \ge \alpha} \in \Sigma$ $\text {(5)}: \quad$ $\ds$  $\ds \forall \alpha \in \R: \set {x \in X: \map f x > \alpha} \in \Sigma$ $\text {(5')}: \quad$ $\ds$  $\ds \forall \alpha \in \Q: \set {x \in X: \map f x > \alpha} \in \Sigma$

## Proof

Each of $(2)$ up to $(5')$ is equivalent to $(1)$ by combining Mapping Measurable iff Measurable on Generator and Generators for Extended Real Sigma-Algebra.

$\blacksquare$