Borel stay-in-a-set games

Consider an n-person stochastic game with Borel state space S, compact metric action sets A 1 , A 2 ,. .. , A n , 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 1 , a 2 ,. .. , a n) of the players. If the payoff to each player i is 1 or 0 according to whether or not the stochastic process of states stays forever in a given Borel set G i , then there is an-equilibrium for every > 0.