nLab Prob

Idea

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.

Definition

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 $B\in\mathcal{B}$,

  $$p\big(f^{-1}(B)\big) \,=\, q(B) \,.$$

The category Prob has

• as objects, probability spaces;

• as morphisms, measure-preserving functions.

Related concepts

• probability space

• Markov kernel

• measurable function

• category of couplings

• categorical probability

• entropy

category: category