A New Class of Locally Convex Spaces with application

TL;DR

This paper introduces the Radon-Nikodym class of locally convex spaces, characterizes it via vector measure derivatives, and applies it to multimeasures and Pettis integrable multifunctions, expanding the understanding of Radon-Nikodym properties.

Contribution

It defines the RN class of locally convex spaces, characterizes it through vector measure derivatives, and extends Radon-Nikodym theorems to multimeasures and Pettis integrable multifunctions.

Findings

RN class properly contains RNP class of complete spaces

Radon-Nikodym theorem extended to multimeasures

Characterization of RN class via integrable derivatives

Abstract

In this paper we define the Radon-Nikodym class (RN class) of locally convex topological vector spaces. The RN class is characterized in terms of the Radon-Nikodym theorem for vector measures using integrable by seminorm derivatives. It is shown that the RNP class of all complete Hausdorff locally convex spaces possessing the Radon-Nikodym property is properly contained in the RN class. As an application we present a Radon-Nikodym theorem for multimeasures with respect to the Pettis integrable multifunctions in terms of the RN class.