Factorization Lemma/Extended Real-Valued Function

Theorem
Let $X$ be a set, and $\left({Y, \Sigma}\right)$ be a measurable space.

Let $f: X \to Y$ be a mapping.

Then an extended real-valued function $g: X \to \overline{\R}$ is $\sigma \left({f}\right)$-measurable iff:


 * There exists a $\Sigma$-measurable mapping $\tilde g: Y \to \overline{\R}$ such that $g = \tilde g \circ f$

where:


 * $\sigma \left({f}\right)$ denotes the $\sigma$-algebra generated by $f$
 * $\overline{\mathcal B}$ denotes the extended real $\sigma$-algebra