论文标题
关于$σ$ - 代数相对于交叉点的产品的分销率的补充和改进
Complements and Improvements Regarding Distributivity of the Product for $σ$-Algebras with Respect to the Intersection
论文作者
论文摘要
我们提供$σ$代数$ \ Mathcal {a} $（在集合$ x $），$ \ MATHCAL {F}，\ MATHCAL {G} $（在集合$ U $上）的各种精制条件（在集合$ x $）上$（\ MATHCAL {a} \ otimes \ Mathcal {f}）\ cap（\ Mathcal {a} \ otimes \ Mathcal {g}）= \ Mathcal {a} a} \ ot \ ot（\ otimes \ left） \\文章将结果概括为Arxiv：2007.06095，并在以前没有涵盖由最数分区生成的$σ$代数的阳性结果。我们还提供了一个证明，只要$ x $不可数，并且在$ x $上存在两个$σ$ -Algebras，它们都可以分开，但它们的交叉点不是。我们介绍了此类结构的例子。在最后一部分中，我们将Arxiv：2007.06095的定理3.3从分析到Blackwell空间的设置。
We present a variety of refined conditions for $σ$ algebras $\mathcal{A}$ (on a set $X$), $\mathcal{F}, \mathcal{G}$ (on a set $U$) such that the distributivity equation $$(\mathcal{A}\otimes\mathcal{F})\cap(\mathcal{A}\otimes\mathcal{G})=\mathcal{A}\otimes\left(\mathcal{F}\cap\mathcal{G}\right),$$ holds -- or is violated. \\ The article generalizes the results in arXiv:2007.06095 and includes a positive result for $σ$ algebras generated by at most countable partitions, was not covered before. We also present a proof that counterexamples may be constructed whenever $X$ is uncountable and there exist two $σ$-algebras on $X$ which are both countably separated, but their intersection is not. We present examples of such structures. In the last section, we extend Theorem 3.3 of arXiv:2007.06095 from analytic to the setting of Blackwell spaces.
