nLab Prob

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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,𝒜,p)(X,\mathcal{A},p) and (Y,,q)(Y,\mathcal{B},q) be probability spaces.

A function f:XYf \colon X\to Y is called

  • measurable if for every BB\in\mathcal{B} we have f 1(B)𝒜f^{-1}(B)\in\mathcal{A};

  • measure-preserving if it is measurable, and moreover for every BB\in\mathcal{B},

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

The category Prob has

category: category

Last revised on January 25, 2024 at 16:34:26. See the history of this page for a list of all contributions to it.