Surjit Singh Khurana
Radon-Nikodym property

Comment.Math.Univ.Carolin. 58,4 (2017) 461-464.

Abstract:For a Banach space $E$ and a probability space $(X, \mathcal{A}, \lambda)$, a new proof is given that a measure $\mu: \mathcal{A} \to E$, with $\mu \ll \lambda$, has RN derivative with respect to $\lambda$ iff there is a compact or a weakly compact $C \subset E$ such that $|\mu |_{C} : \mathcal{A} \to [0, \infty]$ is a finite valued countably additive measure. Here we define $|\mu |_{C}(A) = \sup \{\sum_{k} |\langle \mu (A_{k}), f_{k}\rangle |\}$ where $\{A_{k}\}$ is a finite disjoint collection of elements from~$\mathcal{A}$, each contained in $A$, and $\{f_{k}\}\subset E'$ satisfies $\sup_{k} |f_{k} (C)|\leq 1$. Then the result is extended to the case when $E$ is a Frechet space.

Keywords: liftings; lifting topology; weakly compact sets; Radon-Nikodym derivative

DOI: DOI 10.14712/1213-7243.2015.228
AMS Subject Classification: 46B22 46G05 46G10 28A51 60B05 28B05 28C05