On a Carathbodory-Type Selection Theorem

In the present paper we obtain a Caratheodory-type selection theorem. This result is used in [9] to extend the theory of Nash equilibria, developed in [12, 4, 2, 7, 8, 10, and 13-171, to the setting of an arbitrary measure space of agents and an infinite dimensional strategy space. More specifically, we consider the following problem. Let T be a measure space, Z be a complete separable metric space, and Y be a separable Banach space. Let 4: T x Z + 2’ be a convex valued (possibly empty) correspondence. Let U = { (t, x) E T x Z: &I, x) # a}. Under appropriate assumptions, does there exist a CarathPodory-type selection for 4, i.e., a