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arXiv · 2210.02201

Completion procedures in measure theory

A. G. Smirnov·M. S. Smirnov
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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 $\mu$. With every such ring $\mathcal N$, an extension of $\mu$ is naturally associated which is called the $\mathcal N$-completion of $\mu$. The $\mathcal N$-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 $\sigma$-additivity of a content is preserved under the $\mathcal N$-completion and establish a criterion for the $\mathcal N$-completion of a measure to be again a measure.

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A. G. Smirnov, M. S. Smirnov. 2022-09-25. Completion procedures in measure theory. https://doi.org/10.1007/s10476-023-0233-3

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