# Characterization of Measurable Functions

## Theorem

Let $\left({X, \Sigma}\right)$ be a measurable space.

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

Then the following are all equivalent:

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

## 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.

$\blacksquare$