A factorization of the Giry monad on standard Borel spaces using $[0,\infty]$-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 $\mathcal{V}=[0,\infty]$ which is the one-point compactification of the non-negative real numbers. By viewing probability measures $P \in \mathcal{G}(A)$ as functionals operating on measurable functions $A \rightarrow \mathcal{V}$, and taking the restriction of those functionals to operate on countably affine measurable functions we show that $A \cong Hom_{\mathcal{V}^{\mathcal{V}}}(\mathcal{V}^A,\mathcal{V})$ for all object $A$ lying in a the subcategory $\mathbf{Std}{SCvx}$ of $\mathbf{Std}$. The objects of $\mathbf{Std}{SCvx}$ are standard spaces with a superconvex space structure such that $A$ is coseparable by countably affine measurable functions to $\mathcal{V}$ and satisfies the generic ``fullness property''. The morphisms of the category $\mathbf{Std}{SCvx}$ are countably affine measurable functions. The isomorphism is equivalent to the statement that the full subcategory of $\mathbf{Std}{SCvx}$ consisting of the single object $\mathcal{V}$ is codense in $\mathbf{Std}{SCvx}$ which allows us to easily construct the $\mathcal{G}$-algebras of objects in $\mathbf{Std}{SCvx}$. This permits an adjoint factorization of the Giry monad as the composite of $\mathbf{Std} \xrightarrow{\hat{\mathcal{G}}} \mathbf{Std}{SCvx}$, which is the Giry monad functor viewed as a functor into $\mathbf{Std}{SCvx}$, and the partial forgetful functor $\mathbf{Std}{SCvx} \xrightarrow{\mathcal{U}{SCvx}} \mathbf{Std}$ which forgets the superconvex space structure. We prove that every object in $\mathbf{Std}_{SCvx}$ is an algebra of the $\mathcal{G}$ monad.