Nondentable Sets in Banach Spaces

TL;DR

This paper generalizes Bourgain's result by showing that in any bounded, nondentable set in a Banach space, one can find a separated, weakly closed approximate bush, leading to new examples of divergent quasimartingales.

Contribution

It extends Bourgain's theorem from closed, convex sets to all bounded, nondentable sets, introducing the concept of approximate bushes.

Findings

Existence of separated, weakly closed approximate bushes in nondentable sets

Construction of A-valued quasimartingales with divergent behavior

Generalization of Radon-Nikodým property related structures

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.