On the semimetric on a boolean algebra induced by a finitely additive probability measure

Thomas E. Armstrong, Karel Prikry

Research output: Contribution to journalArticlepeer-review

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Abstract

A finitely additive probability measure μ on a Boolean algebra B induces a semi-metric dμ defined by dμ(A, B) = μ(AΔB). When B is a σ-algebra and μ countably additive B is complete as is well known. The converse is shown to be true. More precisely, if Bμ is the quotient of B via μ-null sets then Bμ is dμ-complete iff μ is countably additive on Bμ and Bμ is complete as a Boolean algebra. Furthermore Bμ is dμ-complete iff every ν≪μ has a Hahn decomposition iff (when B is an algebra of sets) every ν≪μ has B-measurable Radon-Nikodym derivative. If Bμ is not dμ-complete it is either meager in itself or fails to have the property of Baire in it’s completion. Examples are given of both situations with the density character of Bμ an arbitrary infinite cardinal number.

Original languageEnglish (US)
Pages (from-to)249-264
Number of pages16
JournalPacific Journal of Mathematics
Volume99
Issue number2
DOIs
StatePublished - Apr 1982

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