不含无穷大的测度论
Measure theory without infinities
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中文总结 AI 辅助
本文提出一种避免无穷大的测度论框架，通过修改测度概念补充极大性条件，将正测度与向量测度统一处理，改进了Sion和Weber的群值内容扩张结果。
中文摘要 AI 辅助
本文旨在开发一种避免无穷大、可统一处理正测度与向量测度的测度论框架。我们的方法基于对测度概念的修改，在通常的σ-可加性要求之外补充了合适的极大性条件。对每个Hausdorff拓扑向量空间𝔄和σ-环𝒬，我们关联一个对应于𝒬的“无穷”𝔄值测度的向量空间ℳ(𝒬,𝔄)。特别地，ℳ(𝒬,ℝ)的正元素可自然对应于定义在𝒬上的σ-有限正测度，从而将正测度与符号测度置于同一框架中。最后，Sion和Weber提出的群值内容的扩张结果在该新框架内被重新表述并改进。
英文摘要
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.
发表机构
- P. N. Lebedev Physical Institute（列别杰夫物理研究所）
- Lomonosov MSU, Faculty of Computational Mathematics and Cybernetics（莫斯科国立大学计算数学与控制论学院）
机构由 AI 辅助整理，请以论文原文为准。
