Properties of Functions on a Bounded Charge Space

Abstract

Abstract A charge space (X, 𝒜, µ) is a generalisation of a measure space, consisting of a sample space X, a field of subsets 𝒜 and a finitely additive measure µ, also known as a charge. Properties a real-valued function on X may possess include T1-measurability and integrability. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations of T1-measurability and integrability for a bounded charge space (µ(X) < ∞). These characterisations are convenient for analytic purposes; for example, they facilitate simple proofs that T1-measurability is equivalent to conventional measurability and integrability is equivalent to Lebesgue integrability, if (X, 𝒜, µ) is a complete measure space. 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 L1(X, 𝒜, µ) 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 ℒp(X, 𝒜, µ).