The Radon-Nikodým Property for Banach-Spaces — A Survey of Geometric Aspects

Abstract

This chapter analyzes the Radon-Nikodým property (RNP) for Banach-Spaces by using geometric aspects. The definition of the RNP states that a Banach space X (always over the real numbers R) has it if and only if the classical Radon-Nikodým theorem holds for X-valued measures. X has the RNP if and only if for every σ-finite measure space and for every X-valued measure there exists a λ-Bochner integrable function. Furthermore, X has the RNP if and only if for every finite measure space, either of the following equivalent conditions hold true: for every additive there exists bounded measurable function; and for every continuous linear operator there exists a bounded measurable function.