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

2008/03/12 by Tanaka, Jun
#28A12 #28B20 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.0803.1749

Abstract

Let Ω denote an algebra of sets and μ a σ-finite measure. We then prove that the completion of Ω under the pseudometric d(A,B) = μ^∗(A \triangle B) is σ-algebra isomorphic and isometric to the Caratheodory Extension of Ω under the equivalence relation ∼.

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