Basic Principles of Quantum Measure Theory

Abstract

In the present chapter we shall discuss noncommutative analogies of basic principles of measure theory. In every textbook on measure theory one can certainly encounter some of the basic measure-theoretic tools like the Hahn-Jordan and Yosida-Hewitt decompositions of measures, the Boundedness Principle, the Vitali-Hahn-Saks Theorem, the Radon-Nikodým Theorem, the Egoroff Theorem, the Lusin Theorem, and the Lyapunov Theorem. We shall address these principles in the framework of projection lattices instead of Boolean algebras. In the first section we derive the boundedness principle. It says that any completely additive measure on the projection lattice without finite-dimensional component is bounded. This deep result strengthens the Generalized Gleason Theorem for completely additive measures by eliminating the assumption of boundedness. A few consequences, notably the Nikodým boundedness principle for a system of measures, are then discussed. Section 6.2 is devoted to the analysis of continuous and discontinuous parts of finitely additive measures on projections, especially to the Yosida-Hewitt decompositions. The theorems on convergence of sequences of measures, in particular the Vitali-Hahn-Saks theorem, are treated in Section 6.3. In Section 6.4. the historical background is commented on and notes on remaining basic principles (Egoroff Theorem, Lusin Theorem, Radon-Nikodým Theorem, Lyapunov Theorem) are provided.