Random and Vector Measures: from "Toy" Measurable Systems to Quantum Probability

Abstract

Vector measure theory and Bochner integration have been well studied over the past century. This work is an introduction to both theories and explores various examples and applications in each. The theories and theorems are pre-existing, whereas the examples and discussions are mine. Our primary examples of vector measures are toy vector measures, which serve as a class of elementary yet nontrivial structures that enables us to grasp the spirit and essence of the advanced theory, both on the conceptual and technical level. We also discuss random measures as special cases of vector measures. The theory of Bochner integration is introduced as a framework for the Radon-Nikodym Property, which comes from the failure of the Radon-Nikodym Theorem to hold when generalized to Banach spaces. The consequences of this failure as well as Rieffel’s extension of the theorem are discussed in Chapter 2. Finally, we conclude with a brief introduction to Hilbert quantum theory and quantum probability and introduce possible vector extensions of quantum probability theory.