Measurable Selection Theorems for Minimax Stochastic Optimization Problems

Abstract

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.