The category of probabilistic coherence spaces (PCoh_!), introduced by Danos and Ehrhard, is a fully abstract model for PCF with discrete probabilities, where morphisms can be seen as power series. The category Cstab_m, of measurable cones and measurable stable functions, has been introduced by Ehrhard, Pagani and Tasson as a model for PCF with continuous probabilities.

In this talk, we will study the shape of stable functions when they are between discrete cones: we will show that they can actually be seen as generalized power series. The proof is based on a generalization of a theorem from real analysis due to Bernstein, that states that all absolutely monotonous functions on reals are power series. From there, we will build a full and faithful functor from PCoh_! into Cstab_m that moreover preserves the cartesian closed structure.