论文标题
在规范布尔代数的某些拓扑特性上
On some topological properties of normed Boolean algebras
论文作者
论文摘要
本文涉及规范布尔代数（N.B.A.）的紧凑性和可分离性特性，相对于由两个元素之间对称差异的量度等于平方根产生的拓扑。动机是由研究N.B.A.中的随机元素说话值引起的。 这些拓扑特性是重要的假设，使我们能够在概括随机变量收敛的概念，条件定律等的定义时避免可能遇到困难。 对于每个N.B.A.，都有一个有限的测量空间$（e，{\ Mathcal E}，μ）$，使得N.B.A.是$（\ widetilde {\ Mathcal e}，\widetildeμ）$的同构，这是由于初始$σ$ -Algebra的分解而产生的，理想的设置的理想是可忽略的。我们关注$（\ widetilde {\ Mathcal e}，\widetildeμ）$的拓扑属性，当$μ$是无限度量时。如果$μ$是无限的，我们还考虑$ \ widetilde {\ Mathcal E} _ {fin} \ subseteq \ subseteq \ widetilde {\ Mathcal e} $由具有有限度量的可测量集的类别组成的类别。 N.B.A.的紧凑性和可分离性使用相应测量空间的近似性和均匀近似性的新定义术语来表征。 最后，在$（e，\ mathcal e，μ）上的条件是为$ \ widetilde {\ Mathcal e} $和$ \ widetilde {\ Mathcal E} _ {fin}的可分离性和紧凑性而得出的。
This paper concerns the compactness and separability properties of the normed Boolean algebras (N.B.A.) with respect to topology generated by a distance equal to the square root of a measure of symmetric difference between two elements. The motivation arises from studying random elements talking values in N.B.A. Those topological properties are important assumptions that enable us to avoid possible difficulties when generalising concepts of random variable convergence, the definition of conditional law and others. For each N.B.A., there exists a finite measure space $(E, {\mathcal E}, μ)$ such that the N.B.A. is isomorphic to $(\widetilde{\mathcal E}, \widetildeμ)$ resulting from the factorisation of initial $σ$-algebra by the ideal of negligible sets. We focus on topological properties of $(\widetilde{\mathcal E}, \widetildeμ)$ in general setting when $μ$ can be an infinite measure. In case when $μ$ is infinite, we also consider properties of $\widetilde{\mathcal E}_{fin} \subseteq \widetilde{\mathcal E}$ consisting of classes of measurable sets having finite measure. The compactness and separability of the N.B.A. are characterised using the newly defined terms of approximability and uniform approximability of the corresponding measure space. Finally, conditions on $(E,\mathcal E,μ)$ are derived for separability and compactness of $\widetilde{\mathcal E}$ and $\widetilde{\mathcal E}_{fin}.$
