论文标题
非添加度量理论中功能空间的完整性和可分离性
The completeness and separability of function spaces in nonadditive measure theory
论文作者
论文摘要
对于所有可测量函数的空间$ \ MATHCAL {l}^0（μ）$，Choquet-lorentz空间$ \ MATHCAL {l}^{p，q}（μ）$ $ \ Mathcal {l}^\ infty（μ）$ $ $ $ $ $ $ $的可测量函数，其商空间被定义为它们上的合适的前词。在这些函数空间中，Choquet-Lorentz空间由Choquet积分定义，而弱类型的Lorentz空间由Shilkret积分定义。然后研究这些空间的完整性和可分离性。引入了非加工度量的新特征，称为属性（c），以建立$μ$ $ $ $ $ $ $可测量函数的凯奇融合标准。该标准和Choquet和Shilkret积分的合适收敛定理为进行我们的调查提供了工具。
For a nonadditive measure $μ$, the space $\mathcal{L}^0(μ)$ of all measurable functions, the Choquet-Lorentz space $\mathcal{L}^{p,q}(μ)$, the Lorentz space of weak type $\mathcal{L}^{p,\infty}(μ)$, the space $\mathcal{L}^\infty(μ)$ 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.
