On convex hulls of compact sets of probability measures with countable supports

Abstract

E. Michael and I. Namioka proved the following theorem. Let Y be a convex G δ -subset of a Banach space E such that if K ⊂ Y is a compact space, then its closed (in Y) convex hull is also compact. Then every lower semicontinuous set-valued mapping of a paracompact space X to Y with closed (in Y) convex values has a continuous selection. E. Michael asked the question: Is the assumption that Y is G δ essential? In this note we give an affirmative answer to this question of Michael.