The completeness and separability of function spaces in nonadditive measure theory

2022/06/17 by Kawabe, Jun, Yamada, Naoki · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 28E10 #Secondary 46E30

paper · doi:10.48550/arxiv.2206.08511

Abstract

For a nonadditive measure μ, the space L⁰(μ) of all measurable functions, the Choquet-Lorentz space L^p,q(μ), the Lorentz space of weak type L^p,∞(μ), the space L^∞(μ) of all μ-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 μ-measure of measurable functions. This criterion and suitable convergence theorems of the Choquet and Shilkret integrals provide instruments for carrying out our investigation.

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