Properties of functions on a bounded charge space

TL;DR

This paper explores properties of functions on bounded charge spaces, providing new characterisations of measurability and integrability, and extending classical measure theory concepts to this more general setting.

Contribution

It introduces new characterisations of $T_1$-measurability and integrability for bounded charge spaces, and extends the theory of measure space completion to charges.

Findings

New characterisations of $T_1$-measurability and integrability.

Conditions for $L_1$ space to be Banach.

Generalisation of measure space completion for charges.

Abstract

A charge space $(X,\mathcal{A},\mu)$ is a generalisation of a measure space, consisting of a sample space $X$, a field of subsets $\mathcal{A}$ and a finitely additive measure $\mu$, also known as a charge. Key properties a real-valued function on $X$ may possess include $T_1$-measurability and integrability. These properties are generalisations of corresponding properties of real-valued functions on a (countably additive) measure space. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations of $T_1$-measurability and integrability in the case that the charge space is bounded, that is, $\mu(X) < \infty$. These characterisations are convenient for analytic purposes; for example, they facilitate simple proofs that $T_1$-measurability is equivalent to conventional measurability and integrability is equivalent to Lebesgue integrability, if $(X,\mathcal{A},\mu)$ is a complete measure space. Several additional contributions to the theory of bounded charges are also presented. New characterisations of equality almost everywhere of two real-valued functions on a bounded charge space are provided. Necessary and sufficient conditions for the function space $L_1(X,\mathcal{A},\mu)$ to be a Banach space are determined. Lastly, the concept of completion of a measure space is generalised for charge spaces, and it is shown that under certain conditions, completion of a charge space adds no new equivalence classes to the quotient space $\mathcal{L}_p(X,\mathcal{A},\mu)$.