On Convex Hulls of Compact Sets of Probability Measures with Countable Supports

E. Michael and I. Namioka proved the following theorem. Let $Y$ be a convex $G_\delta$-subset of a Banach space $E$ such that if $K\subset Y$ 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 $G_\delta$ essential? In this note we give an affirmative answer to this question of Michael.