We study the validity of the distributivity equation

$$\begin{aligned} ({\mathcal {A}}\otimes {\mathcal {F}})\cap ({\mathcal {A}}\otimes {\mathcal {G}}) ={\mathcal {A}}\otimes \left( {\mathcal {F}}\cap {\mathcal {G}}\right) , \end{aligned}$$

where \({\mathcal {A}}\) is a \(\sigma \)-algebra on a set X, and \({\mathcal {F}}, {\mathcal {G}}\) are \(\sigma \)-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 \(\sigma \)-algebras’ atoms. Using this, we give a sufficient condition under which distributivity holds.

中文翻译：

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$$ \ sigma $$σ-代数相对于交点的乘积（非）分布

我们研究分布方程的有效性

$$ \ begin {aligned}（{\ mathcal {A}} \ otimes {\ mathcal {F}}）\ cap（{\ mathcal {A}} \ otimes {\ mathcal {G}}）= {\ math { A}} \ otimes \ left（{\ mathcal {F}} \ cap {\ mathcal {G}} \ right），\ end {aligned} $$

其中\（{\ mathcal {A}} \）是集合X上的\（\ sigma \）-代数，而\（{\ mathcal {F}}，{\ mathcal {G}} \）是\（ \ sigma \）-集合U上的代数。我们给出了一般情况的反例，并且在可分析可测空间的可数子空间生成的情况下，我们根据\（\ sigma \）-代数的原子给出了一个等价条件。利用这一点，我们给出了分布保持的充分条件。