Affine logic is a fragment of continuous logic introduced by Bagheri, where one only has affine connectives: sum, multiplication by real scalars, and the constant formula 1. This allows to endow the type spaces with the structure of compact convex sets and study the extremal models: the structures that realize only extreme types. Of particular interest are the simplicial theories, all of whose type spaces are Choquet simplices. I will discuss an integral decomposition result for models of simplicial theories into extremal models that generalizes both the ergodic decomposition theorem and the decomposition into factors of tracial von Neumann algebras. We do not make any separability assumptions and the non-separable case of the theorem is new even in these two classical settings. This is joint work with Itaï Ben Yaacov and Tomás Ibarlucía.