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 long-standing open problems and show: (i) any MSE {\it induces} a RSED, and any RSED can be {\it lifted} to a MSE, (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 {\it exact} law of large numbers (ELLN) that can be formalized in the analytic framework of a Fubini extension. We discuss how the \lq measurability' problem associated with a MSE of a large game is automatically resolved in such a framework. We also illustrate our ideas by an approximate result pertaining to a sequence of large but finite games.