Existence of Competitive Equilibria in Markets with a Continuum of Traders

It is well known, and easy to establish, that there exist markets that do not have competitive equilibria, provided the traders do not have convex preferences--that is, that the set of commodity bundles preferred or indifferent to a given bundle is not always convex. It is proved, nevertheless, that in a market consisting of a continuum of traders, each one individually insignificant, there is always a competitive equilibrium, even when the preferences are not convex. (Author)