On some topological properties of normed Boolean algebras

TL;DR

This paper investigates the topological properties of normed Boolean algebras, focusing on compactness and separability, and introduces new concepts like approximability to characterize these properties in measure-theoretic terms.

Contribution

It provides a novel characterization of compactness and separability in normed Boolean algebras using the concepts of approximability and uniform approximability of measure spaces.

Findings

Characterization of compactness via approximability

Conditions for separability in measure spaces

Extension of properties to infinite measure cases

Abstract

This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A. Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others. For each N.B.A., there exists a finite measure space $(E, {\mathcal E}, \mu)$ such that the N.B.A. is isomorphic to $(\widetilde{\mathcal E}, \widetilde{\mu})$ resulting from the factorisation of initial $\sigma$-algebra by the ideal of negligible sets. We focus on topological properties of $(\widetilde{\mathcal E}, \widetilde{\mu})$ in general setting when $\mu$ can be an infinite measure. In case when $\mu$ is infinite, we also consider properties of $\widetilde{\mathcal E}_{fin} \subseteq \widetilde{\mathcal E}$ consisting of classes of measurable sets having finite measure. The compactness and separability of the N.B.A. are characterised using the newly defined terms of approximability and uniform approximability of the corresponding measure space. Finally, conditions on $(E,\mathcal E,\mu)$ are derived for separability and compactness of $\widetilde{\mathcal E}$ and $\widetilde{\mathcal E}_{fin}.$