arXiv subjects

M. S. Smirnov

Publications and source records attributed to M. S. Smirnov.

2 recordsLinked to original sources

arXiv2026

Measure theory without infinities

The aim of this paper is to develop a framework for measure theory that avoids infinities and allows for the uniform treatment of positive and vector measures. Our approach is based on a modification of the notion of measure, which supplements the usual $σ$-additivity requirement with a suitable maximality condition. To each Hausdorff topological vector space $\mathfrak A$ and $σ$-ring $\mathcal{Q}$, we associate a vector space $\mathscr M(\mathcal{Q},\mathfrak A)$ of `infinite' $\mathfrak A$-valued measures corresponding to $\mathcal{Q}$. In particular, the positive elements of $\mathscr M(\mathcal{Q},\mathbb R)$ are naturally identified with the $σ$-finite positive measures defined on $\mathcal{Q}$, thus placing positive and signed measures within the same setting. Finally, extension results for group-valued contents due to Sion and Weber are reformulated and refined within this new framework.

math.FA↗

arXiv2023

Completion procedures in measure theory

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 $\mathcal N$, an extension of $μ$ is naturally associated which is called the $\mathcal N$-completion of $μ$. 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 $σ$-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.

math.FA↗