Factorization Lemma

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Theorem

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

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


Real-Valued Function

Then a mapping $g: X \to \R$ is $\map \sigma f \, / \, \map {\mathcal B} \R$-measurable if and only if:

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

where:

$\map \sigma f$ denotes the $\sigma$-algebra generated by $f$
$\map {\mathcal B} \R$ denotes the Borel $\sigma$-algebra on $\R$


Extended Real-Valued Function

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

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 $f$