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 and be probability spaces.
A function is called
measurable if for every we have ;
measure-preserving if it is measurable, and moreover for every ,
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.