Quantum measure and integration theory

TL;DR

This paper develops a quantum measure and integration theory, introducing a quantum integral that generalizes Lebesgue integration and establishing foundational properties and theorems within quantum measure spaces.

Contribution

It introduces a quantum integral, explores its properties, and demonstrates its relation to classical Lebesgue integration, including a monotone convergence theorem.

Findings

Quantum integral generalizes Lebesgue integral

Monotone convergence theorem established for quantum integrals

Radon-Nikodym theorem does not hold for quantum measures

Abstract

This article begins with a review of quantum measure spaces. Quantum forms and indefinite inner-product spaces are then discussed. The main part of the paper introduces a quantum integral and derives some of its properties. The quantum integral's form for simple functions is characterized and it is shown that the quantum integral generalizes the Lebesgue integral. A bounded, monotone convergence theorem for quantum integrals is obtained and it is shown that a Radon-Nikodym type theorem does not hold for quantum measures. As an example, a quantum-Lebesgue integral on the real line is considered.