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

Tanaka, Jun
Description
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 $\sim$.
Comment: 5 pages
Keywords
Mathematics - Functional Analysis, 28A12, 28B20