nLab
measurable function

Measurable functions

Idea

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

Definitions

Given measurable spaces XX and YY, a measurable function from XX to YY is a function f:XYf\colon X \to Y such that the preimage f *(T)f^*(T) is measurable in XX whenever TT is measurable in YY.

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

In some variations of measure theory based on δ\delta‑ or σ\sigma-rings instead of on σ\sigma-algebras, it is necessary to allow partial functions whose domain is a relatively measurable set. Classically (when YY is the real line), one achieves (for purposes of integration) essentially the same result by requiring only that f 1(I)f^{-1}(I) be measurable whenever IYI \subseteq Y is an interval that does not contain 00; in other words, one effectively assumes that ff 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 XX and YY, then we may allow a measurable function to be an almost function: a partial function whose domain is full. Specifically, an almost function f:XYf\colon X \to Y is measurable if the preimage of every full set in YY is full in XX and the preimage of every measurable set in YY is, if not quite measurable in XX, at least equal to a measurable set in XX up to a full set in XX. (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.

category: analysis

Revised on March 7, 2013 19:56:09 by Zoran Škoda (161.53.130.104)