Approximate equilibrium in pure strategies for non-atomic games

In this paper we establish the existence of a symmetric approximate equilibrium in pure strategies for an anonymous non-atomic game with acontinuum of strategies. This result is a consequence of the fact that in an abstract game with measure valued profiles [as in Schmeidler (1973, Journal of Statistical Physics 7) and Khan (1986, Transactions of the American Mathematical Society 293)] the degenerate best-response profiles are dense in the set of all best- response. We also present a corollary on assignment games with additively separable payoffs and comment on the difficulties with the purification of equilibrium in general assignment games.