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.

$\blacksquare$

Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $8.1$