  We consider multiplayer stochastic games in which the payoff of each player
is a bounded and Borel-measurable function of the infinite play. By using a
generalization of the technique of Martin (1998) and Maitra and Sudderth
(1998), we show four different existence results. In each stochastic game, it
holds for every $\epsilon>0$ that (i) each player has a strategy that
guarantees in each subgame that this player's payoff is at least her maxmin
value up to $\epsilon$, (ii) there exists a strategy profile under which in
each subgame each player's payoff is at least her minmax value up to
$\epsilon$, (iii) the game admits an extensive-form correlated
$\epsilon$-equilibrium, and (iv) there exists a subgame that admits an
$\epsilon$-equilibrium.
