Codensity and the Giry monad

TL;DR

This paper characterizes Giry monads using integration operators and shows they arise as codensity monads from categories of convex sets and affine maps, providing new insights into their structure.

Contribution

It offers a novel description of Giry monads via integration operators and establishes their origin as codensity monads from specific categories.

Findings

Characterization of finitely and countably additive measures via integration operators

Representation of Giry monads as codensity monads

New perspective on Giry monads in categorical terms

Abstract

The Giry monad on the category of measurable spaces sends a space to a space of all probability measures on it. There is also a finitely additive Giry monad in which probability measures are replaced by finitely additive probability measures. We give a characterisation of both finitely and countably additive probability measures in terms of integration operators giving a new description of the Giry monads. This is then used to show that the Giry monads arise as the codensity monads of forgetful functors from certain categories of convex sets and affine maps to the category of measurable spaces.