Carathéodory扩张定理的一个范畴论证明
A categorical proof of the Carathéodory extension theorem
AI总结：
该研究从范畴论视角出发，通过将测度相关概念对应为函子间的变换，构建偏序集取值函子变换扩张的通用框架，利用Kan扩张证明了测度论中的基础结论Carathéodory扩张定理。
摘要：
Carathéodory扩张定理是测度论中的一项基础结论。通常我们并不清楚一般可测子集的具体形态。Carathéodory扩张定理指出，要定义一个测度，我们只需为生成布尔代数中的子集赋值即可。为了从范畴论角度证明该结论，我们用特定的（余）松弛变换与严格变换来表示（预）测度和外测度。此时Carathéodory扩张对应于严格变换的Kan扩张。我们构建了一个用于偏序集取值函子之间变换扩张的通用框架，并给出了若干关于这类变换扩张的存在性与构造方式的结论。随后我们证明了与测度对应的变换和函子满足这些结论，从而证明了Carathéodory扩张定理。
英文摘要：
The Carathéodory extension theorem is a fundamental result in measure theory. Often we do not know what a general measurable subset looks like. The Carathéodory extension theorem states that to define a measure we only need to assign values to subsets in a generating Boolean algebra. To prove this result categorically, we represent (pre)measures and outer measures by certain (co)lax and strict transformations. The Carathéodory extension then corresponds to a Kan extension of strict transformations. We develop a general framework for extensions of transformations between poset-valued functors and give several results on the existence and construction of extensions of these transformations. We proceed by showing that transformations and functors corresponding to measures satisfy these results, which proves the Carathéodory extension theorem.
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