Characterization of Measurable Functions

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


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.