Summary: This paper elucidates the conceptual role that independent randomization plays in non-cooperative game theory. In the context of large (atomless) games in normal form, we present precise formalizations of the notions of a mixed strategy equilibrium (MSE) and of a randomized strategy equilibrium in distributional form (RSED). We offer a resolution of two longstanding open problems and show that (i) any MSE induces a RSED and any RSED can be lifted to a MSE, and (ii) a mixed strategy profile is a MSE if and only if it has the ex post Nash property. Our substantive results are a direct consequence of an exact law of large numbers that can be formalized in the analytic framework of a Fubini extension. We discuss how the “measurability” problem associated with a MSE of a large game is automatically resolved in such a framework. We also present an approximate result pertaining to a sequence of large but finite games.