论文标题
（非）产品的分布$σ$ - 代数相对于交叉
(Non-)Distributivity of the Product for $σ$-Algebras with Respect to the Intersection
论文作者
论文摘要
我们研究分布式方程的有效性$（\ Mathcal {a} \ otimes \ Mathcal {f}）\ cap（\ Mathcal {a} \ otimes \ Mathcal {g}）= \ Mathcal {A} a} \ ot { $ \ MATHCAL {a} $是$ x $上的$σ$ -Algebra，而$ \ Mathcal {f}，\ Mathcal {G} $是$ u $上的$σ$ -Salgebras。我们提出了一般情况的反例，在分析可测量空间的子空间的情况下，我们就$σ$ -SALGEBRAS的原子提供了等效条件。使用此功能，我们提供了足够的条件，在该条件下保持。
We study the validity of the distributivity equation $$(\mathcal{A}\otimes\mathcal{F})\cap(\mathcal{A}\otimes\mathcal{G})=\mathcal{A}\otimes\left(\mathcal{F}\cap\mathcal{G}\right),$$ where $\mathcal{A}$ is a $σ$-algebra on a set $X$, and $\mathcal{F}, \mathcal{G}$ are $σ$-algebras on a set $U$. We present a counterexample for the general case and in the case of countably generated subspaces of analytic measurable spaces we give an equivalent condition in terms of the $σ$-algebras' atoms. Using this, we give a sufficient condition under which distributivity holds.
