  We construct a factorization of the Giry monad through the category of convex
spaces, and show that, provided that no measurable cardinals exist, probability
measures can be viewed as natural transformations. Using the adjunction of this
factorization, we then show the category of Giry algebras is equivalent to the
category of convex measurable spaces where the $\sigma$-algebra structure
associated with a convex space satisfies an elementary property.
