Probabilistic Stable Functions on Discrete Cones are Power Series

We study the category Cstabm of measurable cones and measurable stable functions---a denotational model of an higher-order language with continuous probabilities and full recursion [7]. We look at Cstabm as a model for discrete probabilities, by showing the existence of a cartesian closed, full and faithful functor which embeds probabilistic coherence spaces---a fully abstract denotational model of an higher language with full recursion and discrete probabilities [8]---into Cstabm. The proof is based on a generalization of Bernstein's theorem from real analysis allowing to see stable functions between discrete cones as generalized power series.