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arXiv · 2609.03875

Measure theory without infinities

A. G. Smirnov·M. S. Smirnov
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Abstract

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 $\sigma$-additivity requirement with a suitable maximality condition. To each Hausdorff topological vector space $\mathfrak A$ and $\sigma$-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 $\sigma$-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.

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A. G. Smirnov, M. S. Smirnov. 2026-09-03. Measure theory without infinities. https://arxiv.org/abs/2609.03875

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