Regularity properties of distributions of correspondences without countable generation: applications to large games

We show that each of the regularity properties of regular conditional distributions of correspondences (convexity, closedness, compactness, and preservation of closed graphs) is equivalent to the condition of nowhere equivalence. This result does not require any countable-generation assumptions. As an application, we establish the existence of a pure-strategy equilibrium for large games with general trait spaces. The trait space may be an arbitrary measurable space. As a corollary, we obtain the existence of a pure-strategy equilibrium in semi-anonymous settings in which payoffs depend, in addition to agents' own actions, on the joint distribution over the space of agents and actions.