Improved conditions for the distributivity of the product for σ-algebras with respect to the intersection

We present a variety of refined conditions for -algebras 𝓐 (on a set ), 𝓕, 𝓖 (on a set ) such that the distributivity equation holds – or is violated. The article generalizes the results in an article of Steinicke (2021) and includes a positive result for -algebras generated by at most countable par...

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Bibliographic Details
Published inMathematica Slovaca Vol. 74; no. 2; pp. 331 - 338
Main Authors Rao, K. P. S. Bhaskara, Steinicke, Alexander
Format Journal Article
LanguageEnglish
Published De Gruyter 25.04.2024
Subjects
03E15
counterexample for sigma-algebras
intersection of sigma-algebras
Primary 28A05
product sigma-algebras
Secondary 60A05
sigma-algebra
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Summary:We present a variety of refined conditions for -algebras 𝓐 (on a set ), 𝓕, 𝓖 (on a set ) such that the distributivity equation holds – or is violated. The article generalizes the results in an article of Steinicke (2021) and includes a positive result for -algebras generated by at most countable partitions, which was not covered before. We also present a proof that counterexamples may be constructed whenever is uncountable and there exist two -algebras on which are both countably separated, but their intersection is not. We present examples of such structures. In the last section, we extend Theorem 2 in Steinicke (2021) from analytic to the setting of Blackwell spaces.
ISSN:0139-9918
1337-2211
DOI:10.1515/ms-2024-0025