Measure theory without infinities
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 σ-additivity requirement with a suitable maximality condition. To each Hausdorff topological vector space A and σ-ring 𝒬, we associate a vector space ℳ(𝒬,A) of `infinite' A-valued measures corresponding to 𝒬. In particular, the positive elements of ℳ(𝒬,ℝ) are naturally identified with the σ-finite positive measures defined on 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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BibTeX
@misc{smirnov2026measure,
title = {{Measure theory without infinities}},
author = {A. G. Smirnov and M. S. Smirnov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03875},
eprint = {2609.03875},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03875},
}
RIS
TY - UNPB
TI - Measure theory without infinities
AU - A. G. Smirnov
AU - M. S. Smirnov
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03875
UR - https://arxiv.org/abs/2609.03875
AB - 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 A and σ-ring 𝒬, we associate a vector space ℳ(𝒬,A) of `infinite' A-valued measures corresponding to 𝒬. In particular, the positive elements of ℳ(𝒬,ℝ) are naturally identified with the σ-finite positive measures defined on 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.
ER -