An isomorphism between the completion of an Algebra and its Caratheodory Extension

TL;DR

This paper establishes an isomorphism between the completion of an algebra of sets under a measure-induced pseudometric and its Carathéodory extension, providing a new perspective on measure-theoretic completions.

Contribution

It proves that the completion of an algebra of sets under a specific pseudometric is isomorphic and isometric to its Carathéodory extension, linking two fundamental measure-theoretic constructions.

Findings

The completion under the measure-induced pseudometric is a σ-algebra.

The completion is isomorphic to the Carathéodory extension.

The completion preserves the measure-theoretic structure.

Abstract

Let $\Omega$ denote an algebra of sets and $\mu$ a $\sigma$-finite measure. We then prove that the completion of $\Omega$ under the pseudometric $d(A,B)$ = $\mu^{\ast}(A \triangle B)$ is $\sigma$-algebra isomorphic and isometric to the Caratheodory Extension of $\Omega$ under the equivalence relation $\sim$.