A General Theory of Operator-Valued Measures

TL;DR

This paper introduces projection families, a new class of measures generalizing operator-valued measures, enabling integrals as elements of the second dual and satisfying key convergence theorems.

Contribution

It develops a new measure concept called projection families, broadening the scope of operator-valued measures with easier-to-construct properties.

Findings

Projection families generalize classical operator-valued measures.

They admit integrals as elements of the second dual space.

Projection families satisfy Monotone and Dominated Convergence theorems.

Abstract

We construct a new kind of measures, called projection families, which generalize the classical notion of vector and operator-valued measures. The maximal class of reasonable functions admits an integral with respect to a projection family, where the integral is defined as an element of the second dual instead of the original space. We show that projection families possess strong enough properties to satisfy the theorems of Monotone Convergence and Dominated convergence, but are much easier to come by than the more restrictive operator-valued measures.