Deriving the Giry algebras on standard Borel spaces using $\mathbb{R}_{\infty}$-generalized points

Abstract

The Giry monad on the category of measurable spaces restricts to the full subcategory of standard Borel spaces, $\mathbf{Std}$, which we show is amenable to analysis. $\mathbf{Std}$ contains the space $\mathbb{R}_{\infty}$ which is the one-point compactification of the real numbers. By viewing probability measures $P \in \mathcal{G}(A)$ as functionals operating on measurable functions $A \rightarrow \mathbb{R}_{\infty}$, and taking the restriction of those functionals to operate on affine measurable functions we show that $A \cong Hom_{\mathbb{R}_{\infty}^{\mathbb{R}_{\infty}}}(\mathbb{R}_{\infty}^A|,\mathbb{R}_{\infty})$ for all object $A$ lying in the subcategory $\mathbf{Std}_{Cvx}$ of $\mathbf{Std}$. The objects of $\mathbf{Std}_{Cvx}$ are standard spaces with a convex space structure which satisfies the generic ``fullness property''. The morphisms of the category $\mathbf{Std}_{Cvx}$ are affine measurable functions. The isomorphism is equivalent to the statement that the full subcategory of $\mathbf{Std}_{Cvx}$ consisting of the single object $\mathbb{R}_{\infty}$ is codense in $\mathbf{Std}_{Cvx}$ which allows us to easily construct the $\mathcal{G}$-algebras of objects in $\mathbf{Std}_{Cvx}$. This permits an adjoint factorization of the Giry monad as the composite of $\mathbf{Std} \xrightarrow{\hat{\mathcal{G}}} \mathbf{Std}_{Cvx}$, which is the Giry monad functor viewed as a functor into $\mathbf{Std}_{Cvx}$, and the partial forgetful functor $\mathbf{Std}_{Cvx} \xrightarrow{\mathcal{U}_{Cvx}} \mathbf{Std}$ which forgets the convex space structure. We prove that the category $\mathbf{Std}_{Cvx}$ is the category of algebras of the $\mathcal{G}$-monad.