发表机构
University of Connecticut（康涅狄格大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文重新审视 Carathéodory 方法 I，给出忠实扩展的准则，并利用该准则在拓扑空间中简化条件，进而证明多个经典测度论结果，如 Tonelli 定理、Riesz 表示和 Frostman 引理。
AI 中文摘要
Carathéodory 构造，也称为方法 I，将集合族上的任意权重转化为外测度。难点在于扩展问题：证明外测度保留给定的权重。我们记录了方法 I 的基本性质，首先指出覆盖族上的可数次可加性正是忠实扩展的准则，并利用这些性质仅用外测度、σ-代数和测度（无需代数上的预测度）证明标准结果。在拓扑空间中，连续性原理将此准则简化为有限次可加性以及用开集和紧集的有限逼近。应用包括 Lebesgue 积分作为测度、Tonelli 定理、多维 Lebesgue–Stieltjes 测度、向量值泛函的 Riesz 表示、Kolmogorov 扩展、嵌套划分上的质量分布以及 Frostman 引理。
英文摘要
Carathéodory's construction, also known as Method I, turns any weight on a family of sets into an outer measure. The difficulty is the extension problem: showing that the outer measure retains the prescribed weights. We record elementary properties of Method I, beginning with the fact that countable subadditivity on the covering family is exactly the criterion for faithful extension, and use them to prove standard results with only outer measures, $σ$-algebras, and measures, without premeasures on algebras. In topological spaces, a continuity principle reduces this criterion to finite subadditivity and finite approximation by open and compact sets. Applications include the Lebesgue integral as a measure, Tonelli's theorem, multidimensional Lebesgue--Stieltjes measures, Riesz representation for vector-valued functionals, Kolmogorov extension, mass distributions on nested partitions, and Frostman's lemma.
Comments31 pages, comments welcome
