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

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

2008-03-12

arxiv.org/abs/0803.1749v1

Mathematics

Functional Analysis

5 pages

Scientific paper

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 $\sim$.

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