A General Theory of Operator-Valued Measures

TL;DR

The paper introduces projection families as a unifying generalization of vector and operator-valued measures, yielding integrals that take values in the second dual $X^{**}$. It demonstrates that projection families satisfy Monotone Convergence and Dominated Convergence theorems and have finite semivariation, while being more readily constructed than traditional operator-valued measures. By connecting projection families to vector and operator measures and extending Lewis integration, the work provides a framework that encompasses existing integration theories and points toward a Spectral Theorem for Banach-space operators via projection families. The concrete examples in Banach and Hilbert spaces, spectral measures, and quantum-measurement-like constructs illustrate the breadth of the approach and its potential for functional calculus beyond Hilbert spaces.

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.

Theorems & Definitions (60)

• Definition
• Definition
• Definition
• Proposition 2.0.1
• proof
• Corollary 2.0.1: Monotone Convergence Theorem
• proof
• Corollary 2.0.2: Dominated Convergence Theorem
• proof
• Definition