We prove that a two-person, zero-sum stochastic game with arbitrary state and action spaces, a finitely additive law of motion and a bounded Borel measurable payoff has a value.

Consider an n-person stochastic game with Borel state space S, compact metric action sets A <Subscript>1</Subscript>,A <Subscript>2</Subscript>,…,A <Subscript> n </Subscript>, and law of motion q such that the integral under q of every bounded Borel measurable function depends measurably on the initial state x and continuously on the actions (a <Subscript>1</Subscript>,a <Subscript>2</Subscript>,…,a <Subscript> n </Subscript>)...</subscript></subscript></subscript></subscript></subscript></subscript>