On the Radon-Nikodym-Property, and related topics in locally convex spaces

Abstract

We introduce L X 1 (μ), the space of classes of X-valued μ-integrable functions used by Saab, which is an extension of the space of classes of Bochner-integrable functions, in Banach spaces. X denotes here a sequentially complete locally convex space.

We give examples of spaces which are dentable, σ-dentable, having the Radon-Nikodym-Property, or having the Bishop-Phelps-Property, by proving some projective limit results.

We also prove the following theorem : The following implications are valid : TeX

1.X has the Radon-Nikodym-Property. 2.Every uniformly bounded martingale is L X 1 -convergent. 3.Every uniformly bounded martingale is L X 1 -Cauchy. 4.Every uniformly bounded and finitely generated martingale is L X 1 -Cauchy. 5.X is σ-dentable.

So we have the equivalency of (i) through (v) for quasi-complete (BM)-spaces.