Factorization Lemma/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 a mapping $g: X \to \R$ is $\sigma \left({f}\right) \, / \, \mathcal B \left({\R}\right)$-measurable iff:


 * There exists a $\Sigma \, / \, \mathcal B \left({\R}\right)$-measurable mapping $\tilde g: Y \to \R$ such that $g = \tilde g \circ f$

where:


 * $\sigma \left({f}\right)$ denotes the $\sigma$-algebra generated by $f$
 * $\mathcal B \left({\R}\right)$ denotes the Borel $\sigma$-algebra on $\R$