Measurable functions

Idea

The measurable functions are those functions between measure spaces amenable to treatment in measure theory.

Definitions

Given measurable spaces $X$ and $Y$, a measurable function from $X$ to $Y$ is a function $f:X\to Y$ such that the preimage $f^{*}(T)$ is measurable in $X$ whenever $T$ is measurable in $Y$.

In classical measure theory, it is usually assumed that $Y$ is the real line (or a variation such as the extended real line or the complex plane) equipped with the Borel sets. Then $f$ is measurable if and only if $f^{-1}(I)$ is measurable whenever $I\subseteq Y$ is an interval. More generally, if $Y$ is any topological space equipped with the Borel sets, then $f$ is measurable if and only if $f^{-1}(I)$ is measurable whenever $I\subseteq Y$ is open.

In some variations of measure theory based on $\delta$‑ or $\sigma$-rings instead of on $\sigma$-algebras, it is necessary to allow partial functions whose domain is a relatively measurable set. Classically (when $Y$ is the real line), one achieves (for purposes of integration) essentially the same result by requiring only that $f^{-1}(I)$ be measurable whenever $I\subseteq Y$ is an interval that does not contain $0$; in other words, one effectively assumes that $f$ is zero wherever it would otherwise be undefined.

Modulo null sets

If (as in a measure space, a Cheng space, or a localisable measurable space), we have a notion of null sets (or full sets) in $X$ and $Y$, then we may allow a measurable function to be an almost function: a partial function whose domain is full. Specifically, an almost function $f:X\to Y$ is measurable if the preimage of every full set in $Y$ is full in $X$ and the preimage of every measurable set in $Y$ is, if not quite measurable in $X$, at least equal to a measurable set in $X$ up to a full set in $X$. (To emphasise this last change, we may call such functions almost measurable.) Additionally, we consider two measurable almost functions to be equal (or equivalent if one prefers) if they are almost equal: their equaliser is full.