(Non-)Distributivity of the Product for σ-Algebras with Respect to the Intersection

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We study the validity of the distributivity equation (A⊗F)∩(A⊗G)=A⊗(F∩G),where A is a σ-algebra on a set X, and F, 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.

Original languageEnglish
Pages (from-to)667-675
Number of pages9
JournalArchiv der Mathematik
Issue number6
Publication statusPublished - Jun 2021

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