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 $ε>0$ 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 $ε$, (ii) there exists a strategy profile under which in each subgame each player's payoff is at least her minmax value up to $ε$, (iii) the game admits an extensive-form correlated $ε$-equilibrium, and (iv) there exists a subgame that admits an $ε$-equilibrium.