  We prove some regularity properties (convexity, closedness, compactness and
preservation of upper hemicontinuity) for distribution and regular conditional
distribution of correspondences under the nowhere equivalence condition. We
show the necessity of such a condition for any of these properties to hold. As
an application, we demonstrate that the nowhere equivalence condition is
satisfied on the underlying agent space if and only if pure-strategy Nash
equilibria exist in general large games with any fixed uncountable compact
action space.
