Measurable Selection Theorems for Minimax Stochastic Optimization Problems

This paper resolves the measurability questions which arise in the analysis of various minimax stochastic optimization models posed in Borel spaces. For a large class of such models, we provide a set of sufficient conditions which ensure that the questions have positive answers. These conditions are essentially weaker than those described in the existing literature. Moreover, they allow us to conclude a fundamental theorem yielding some new results on the existence of universally measurable selections of extrema for both zero-sum stochastic games and minimax stochastic optimal control problems. In particular, a random version of the well-known minimax theorem of Fan is in this way established. The fundamental theorem yields also a descriptive set theoretic fact concerning the projections of Borel sets. The paper indicates also some counterexamples to possibly more general problems.