Existence of equilibria in repeated games with long-run payoffs

We consider repeated games with tail-measurable payoffs, i.e., when the payoffs depend only on what happens in the long run. We show that every repeated game with tail-measurable payoffs admits an ε-equilibrium, for every ε > 0, provided that the set of players is finite or countably infinite and the action sets are finite. The proof relies on techniques from stochastic games and from alternating-move games with Borel-measurable payoffs.