The algebras of the Giry monad

Abstract

The algebras of the Giry monad on the category of measurable spaces are derived. These algebras, which we denote as ${\mathbf{Cvx}}_{Meas}$, are measurable convex spaces satisfying three properties: (1) all the affine sum operations are measurable functions, (2) $X$ is coseparated by affine measurable maps to ${\mathbb{R}}_{\infty }$, and (3) $X$ satisfies a fullness property. The category of algebras has as morphisms affine measurable functions.

arXiv comment11 pages. This document is an extended version of the post on nLab for the Giry monad