  We present some regularity properties (convexity, weak/weak* compactness and
preservation of weak/weak* upper hemicontinuity) for Bochner/Gelfand
conditional expectation of Banach valued correspondences under the nowhere
equivalence condition. These regularity properties for Bochner/Gelfand integral
of Banach valued correspondences are obtained as corollaries. Similar
properties for regular conditional distributions are also covered by the
corresponding results for Gelfand conditional expectation of correspondences.
We prove the necessity of the nowhere equivalence condition for any of these
properties to hold. As an application, we show that the nowhere equivalence
condition is satisfied on the space of players if and only if a pure-strategy
Nash equilibrium exists in a general class of large games.

Probability