Nash equilibrium and the law of large numbers

Pascoa showed that the failure of the law of large numbers for a continuum of independent randomizations implies that Schmeidler's concept of a measure-valued profile function in equilibrium might not coincide with the concept of mixed strategies equilibrium of a nonatomic game. The latter should be defined as a probability measure on pure strategies profiles which is induced by the product measure of players' mixed strategies. This paper addresses existence and approximate purification of the latter and presents an assumption on continuity of payoffs that guarantees the equivalence between the two equilibrium concepts.