Probability monads as codensity monads

TL;DR

This paper explores how probability measures on various spaces can be understood as extensions of distributions on countable sets, using categorical constructions called codensity monads, and provides integral representation theorems for these measures.

Contribution

It introduces a categorical framework for probability measures as codensity monads, unifying various classes of measures through new constructions and integral representation theorems.

Findings

Probability measures arise as codensity monads of functors from countable sets.

Constructs probability monads for (pre)measures, Radon measures, and Baire measures.

Provides generalized Daniell-Stone theorems for integral representations.

Abstract

We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads as the codensity monads of functors that send a countable set to the space of probability distributions on that set. On (pre)measurable spaces we discuss monads of probability (pre)measures and their finitely additive analogues. We also give codensity constructions for monads of Radon measures on compact Hausdorff spaces and compact metric spaces and for the monad of Baire measures on Hausdorff spaces. A crucial role in these constructions is given by integral representation theorems, which we derive from a generalized Daniell-Stone theorem.