The completeness and separability of function spaces in nonadditive measure theory

TL;DR

This paper investigates the mathematical properties, specifically completeness and separability, of various function spaces built over nonadditive measures, introducing a new measure property to facilitate convergence analysis.

Contribution

It introduces property (C) for nonadditive measures and applies it to establish convergence criteria, advancing the understanding of function space properties in nonadditive measure theory.

Findings

Proves completeness of Choquet-Lorentz and related spaces.

Establishes separability conditions for these function spaces.

Introduces property (C) to characterize measure convergence.

Abstract

For a nonadditive measure $\mu$, the space $\mathcal{L}^0(\mu)$ of all measurable functions, the Choquet-Lorentz space $\mathcal{L}^{p,q}(\mu)$, the Lorentz space of weak type $\mathcal{L}^{p,\infty}(\mu)$, the space $\mathcal{L}^\infty(\mu)$ of all $\mu$-essentially bounded measurable functions, and their quotient spaces are defined together with suitable prenorms on them. Among those function spaces, the Choquet-Lorentz space is defined by the Choquet integral, while the Lorentz space of weak type is defined by the Shilkret integral. Then the completeness and separability of those spaces are investigated. A new characteristic of nonadditive measures, called property (C), is introduced to establish the Cauchy criterion for convergence in $\mu$-measure of measurable functions. This criterion and suitable convergence theorems of the Choquet and Shilkret integrals provide instruments for carrying out our investigation.