Abstract
We propose a unified treatment of extensions of group-valued contents (i.e., additive set functions defined on a ring) by means of adding new null sets. Our approach is based on the notion of a completion ring for a content μ. With every such ring \({\cal N}\), an extension of μ is naturally associated which is called the \({\cal N}\)-completion of μ. The \({\cal N}\)-completion operation comprises most previously known completion-type procedures and also gives rise to some new extensions, which may be useful for constructing counterexamples in measure theory. We find a condition ensuring that σ-additivity of a content is preserved under the \({\cal N}\)-completion and establish a criterion for the \({\cal N}\)-completion of a measure to be again a measure.
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M.S. Smirnov was supported by the Moscow Center of Fundamental and Applied Mathematics at INM RAS (Agreement with the Ministry of Education and Science of the Russian Federation No. 075-15-2022-286).
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Smirnov, A.G., Smirnov, M.S. Completion Procedures in Measure Theory. Anal Math 49, 855–880 (2023). https://doi.org/10.1007/s10476-023-0233-3
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DOI: https://doi.org/10.1007/s10476-023-0233-3