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