Characterization of Measurable Functions

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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