This paper deals with zero-sum discrete-time stationary models of stochastic games with Borel state and action spaces. It is known that the value function of a Borel space stochastic game which satisfies neither semi-continuity nor compactness conditions need not be universally measurable [U. Rieder, On semi-continuous dynamic games, Tech. Rep., Univ. Karlsruhe (1978)]. The main purpose of this paper is to give some semi-continuity and compactness conditions that ensure the existence and universal measurability of the value function of the game as well as the existence of universally measurable optimal strategies for both players. The fundamental result in this paper is a minimax selection theorem extending a selection theorem of L. D. Brown and R. Purves [Ann. Statist. 1, 902-912 (1973; Zbl 0265.28003)]. As applications of this basic result, some new theorems on absorbing discounted, and positive stochastic games are proved.