Year: 2026
Type: preprint
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 $\mathfrak A$ 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.
Source: arXiv (Cornell University)
Authors: Smirnov, A. G., Smirnov, M. S.
Institution: Lomonosov Moscow State University
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208783686