Characterization of Measurable Functions

Theorem
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:

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.