Some characterizations of weak Radon-Nikodým sets

Abstract:

Let $K$ be a weak*-compact convex subset of the dual ${E^*}$ of a Banach space $E$. It is shown that $K$ has the weak Radon-Nikodym property if and only if every ${x^{**}}$ in ${E^{**}}$ restricted to $K$ is universally measurable if and only if every ${x^{**}}$ in ${E^{**}}$ restricted to any weak*-compact subset $M$ of $K$ has a point of continuity on ($M$, weak*) if and only if $K$ is a set of complete continuity if and only if every subset of $K$ is weak* dentable in $(M,\sigma ({E^*},{E^{**}}))$.