Duality and free measures in vector spaces, the spectral theory of actions of non-locally compact groups

TL;DR

This paper develops a duality theory for vector measure spaces, linking measure geometry with measurable linear functionals, introducing the concept of free measures and exploring their applications.

Contribution

It introduces a general duality framework for vector measure spaces and the novel concept of free measures, expanding the understanding of measure geometry and functional analysis.

Findings

Establishes a correspondence between measure geometry and measurable linear functionals.

Introduces the notion of free measures and demonstrates their usefulness.

Provides applications of the duality theory in vector spaces.

Abstract

The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and the properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications.