(Non-)Distributivity of the Product for $\sigma$-Algebras with Respect to the Intersection
Arch. Math. (2021) 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 $\sigma$-algebra on a set $X$, and $\mathcal{F}, \mathcal{G}$ are $\sigma...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
12.07.2020
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Subjects | |
Online Access | Get full text |
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Summary: | Arch. Math. (2021) 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 $\sigma$-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. |
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DOI: | 10.48550/arxiv.2007.06095 |