Factorization Lemma

Theorem

Let $X$ be a set.

Let $\struct {Y, \Sigma}$ be a measurable space.

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

Real-Valued Function

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

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

where:

$\map \sigma f$ denotes the $\sigma$-algebra generated by $f$

$\map \BB \R$ denotes the Borel $\sigma$-algebra on $\R$

Extended Real-Valued Function

An extended real-valued function $g: X \to \overline \R$ is $\map \sigma f$-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:

$\map \sigma f$ denotes the $\sigma$-algebra generated by $f$

This article is complete as far as it goes, but it could do with expansion. In particular: I'd be surprised if this doesn't hold for arbitrary $\sigma$-algebras Not sure what you expect but if we consider $\set {\empty, \R}$ instead of $\map {\mathcal B} \R$, then $f : X \to Y$ and $g : X \to \R$ can be arbitrary mappings. Choosing $f$ to be constant, but $g$, there seems no $\tilde g$ You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.