We extend Aumann's celebrated Walrasian existence result for a pure exchange economy with a measure space of consumers (1966) to the situation where the preferences of a consumer are allowed to depend upon the consumption choices of the other consumers. For economies with finitely many consumers such an extension is rather simple to give; it goes back to Arrow and Debreu (1954). However, for economies with a continuum of players this kind of extension was a long-standing open problem. This is because great topological difficulties manifest themselves if one tries to apply fixed point theorems in the standard way. Mathematically, our solution of the problem turns around a so-called relaxation of the equilibrium existence problem. This introduces (artificial) randomization and thereby allows application of weak convergence results. On the economical side, we also introduce a new notion of asymptotic competitive equilibrium and show that it naturally complements Aumann's original equilibrium notion.