A function between measure- or probability spaces is called measure-preserving if its preimages preserve the measures of subsets.
Such measure-preserving functions are closed under composition, hence they constitute the morphisms of a category whose objects are measure spaces. When restricted to objects that are probability spaces, the category is usually denoted by Prob.
Let $(X,\mathcal{A},p)$ and $(Y,\mathcal{B},q)$ be probability spaces.
A function $f \colon X\to Y$ is called
measurable if for every $B\in\mathcal{B}$ we have $f^{-1}(B)\in\mathcal{A}$;
measure-preserving if it is measurable, and moreover for every $B\in\mathcal{B}$,
The category Prob has
as morphisms, measure-preserving functions.
Last revised on January 25, 2024 at 16:34:26. See the history of this page for a list of all contributions to it.