Nondentable Sets in Banach Spaces

Abstract

In his study of the Radon Nikod\'ym property of Banach spaces, Bourgain showed (among other things) that in any closed, bounded, convex set $A$ that is nondentable, one can find a separated, weakly closed bush. In this note, we prove a generalization of Bourgain's result: in any bounded, nondentable set $A$ (not necessarily closed or convex) one can find a separated, weakly closed approximate bush. Similarly, we obtain as corollaries the existence of $A$-valued quasimartingales with sharply divergent behavior.