论文标题
carathéodory扩展定理的分类证明
A categorical proof of the Carathéodory extension theorem
论文作者
论文摘要
carathéodory扩展定理是度量理论的基本结果。通常，我们不知道一般可测量的子集是什么样。 Carathéodory扩展定理指出，要定义一个度量，我们只需要将值分配给生成的布尔代数中的子集即可。 为了明确证明这一结果，我们通过某些（CO）LAX和严格的转换表示（预先）度量和外部度量。然后，Carathéodory扩展对应于严格转换的KAN扩展。我们开发了一个通用框架，用于扩展Poset值函数之间的转换，并为这些转换的扩展的存在和构建提供了几个结果。我们通过证明与度量相对应的转换和函子可以满足这些结果的进行进行，这证明了Carathéodory扩展定理。
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.
