Jun Kawabe | Searcharxiv

arXiv2022

Topological and topological linear properties of the Sugeno-Lorentz spaces

The properties of the spaces of Sugeno integrable functions are quite different from those of the ordinary spaces of Lebesgue integrable functions. The purpose of the paper is to further advance our study of the Sugeno-Lorentz spaces, in which the completeness and separability of the spaces were discussed. As a continuation of our previous research, in this paper, some fundamental topological and topological linear properties of the Sugeno-Lorentz spaces are investigated in detail.

math.CA↗

arXiv2022

The completeness and separability of function spaces in nonadditive measure theory

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.

math.CA↗