A categorical proof of the Carath\'eodory extension theorem

TL;DR

This paper provides a categorical proof of the Carathéodory extension theorem by representing measures as transformations and using Kan extensions within a general framework for extending transformations.

Contribution

It introduces a categorical framework for measure extension theorems, offering a new proof of Carathéodory's theorem through Kan extensions of transformations.

Findings

Categorical framework for measure extensions developed.

Proof of Carathéodory theorem via Kan extensions.

General results on extending transformations between poset-valued functors.

Abstract

The Carath\'eodory extension theorem is a fundamental result in measure theory. Often we do not know what a general measurable subset looks like. The Carath\'eodory 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 Carath\'eodory 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 Carath\'eodory extension theorem.