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

The Giry monad on the category of measurable spaces restricts to the full subcategory of standard Borel spaces, $\mathbf{S}\mathbf{t}\mathbf{d}$, which we show is amenable to analysis. $\mathbf{S}\mathbf{t}\mathbf{d}$ contains the space ${\mathbb{R}}_{\mathrm{\infty }}$ 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 $A\to {\mathbb{R}}_{\mathrm{\infty }}$, and taking the restriction of those functionals to operate on affine measurable functions we show that $A\cong Ho{m}_{{\mathbb{R}}_{\mathrm{\infty }}^{{\mathbb{R}}_{\mathrm{\infty }}}}({\mathbb{R}}_{\mathrm{\infty }}^A\mathrm{\mid },{\mathbb{R}}_{\mathrm{\infty }})$ for all object $A$ lying in the subcategory ${\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}$ of $\mathbf{S}\mathbf{t}\mathbf{d}$. The objects of ${\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}$ are standard spaces with a convex space structure which satisfies the generic ``fullness property’'. The morphisms of the category ${\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}$ are affine measurable functions. The isomorphism is equivalent to the statement that the full subcategory of ${\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}$ consisting of the single object ${\mathbb{R}}_{\mathrm{\infty }}$ is codense in ${\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}$ which allows us to easily construct the $\mathcal{G}$-algebras of objects in ${\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}$. This permits an adjoint factorization of the Giry monad as the composite of $\mathbf{S}\mathbf{t}\mathbf{d}\stackrel{\widehat{\mathcal{G}}}{\to }{\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}$, which is the Giry monad functor viewed as a functor into ${\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}$, and the partial forgetful functor ${\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}\stackrel{{\mathcal{U}}_{Cvx}}{\to }\mathbf{S}\mathbf{t}\mathbf{d}$ which forgets the convex space structure. We prove that the category ${\mathbf{S}\mathbf{t}\mathbf{d}}_{Cvx}$ is the category of algebras of the $\mathcal{G}$-monad.