On ws-convergence of product measures

A number of fundamental results, centered around extensions of Prohorov's theorem, is proven for the ws-topology for measures on a product space. These results contribute to the foundations of stochastic decision theory. They also subsume the principal results of Young measure theory, which only considers product measures with a fixed, common marginal. Specializations yield the criterion for relative ws-compactness of Schäl, the refined characterizations of ws-convergence of Galdéano and Truffert, and a new version of Fatou's lemma in several dimensions. In a separate, nonsequential development, a generalization is given of the relative ws-compactness criterion of Jacod and Mémin. New applications are given to the existence of optimal equilibrium distributions over player-action pairs in game theory and the existence of most optimistic scenarios in stochastic decision theory.