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Probability Algebras and Ultraproducts

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Abstract

The article discusses the so-called probability algebras and their hermitian representations. A states, quasiinvariant with respect to the hermitian representation of a probability algebra, are considered. The concept of contiguity of sequences of faithful states, which is a generalization of the concept of absolute continuity of states, is also studied. Conditional expectations with respect to subalgebras of probability algebras and their properties are also considered. The ultraproducts of probability algebras is introduced, their properties are studied. It is shown that if the sequences of states are contiguous, then the ultraproducts of this sequences are absolutely continuous. The conditions, under which a conditional expectation on a ultraproduct of sequence of probability algebras exists, are given.

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Lobachevskii Institute of Mathematics and Mechanics, Kazan Federal University, 420008, Kazan, Tatarstan, Russia

    S. G. Haliullin

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  1. S. G. Haliullin
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Correspondence to S. G. Haliullin.

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Haliullin, S.G. Probability Algebras and Ultraproducts. Lobachevskii J Math 45, 412–415 (2024). https://doi.org/10.1134/S1995080224010207

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  • DOI: https://doi.org/10.1134/S1995080224010207

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