The completeness and separability of function spaces in nonadditive measure theory

For a nonadditive measure mu, the space L0(mu) of all measurable functions, the Choquet-Lorentz space Lp,q (mu), the Lorentz space of weak type Lp,degrees O(mu), the space L degrees O(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. (c) 2022 Elsevier B.V. All rights reserved.