Vol. 99, No. 2, 1982

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

Thomas E. Armstrong and Karel Libor Prikry

Vol. 99 (1982), No. 2, 249–264

A finitely additive probability measure μ on a Boolean algebra induces a semi-metric dμ defined by dμ(A,B) = μ(AΔB). When is a σ-algebra and μ countably additive is complete as is well known. The converse is shown to be true. More precisely, if μ is the quotient of via μ-null sets then μ is dμ-complete iff μ is countably additive on μ and μ is complete as a Boolean algebra. Furthermore μ is dμ-complete iff every ν μ has a Hahn decomposition iff (when is an algebra of sets) every ν μ has a -measurable Radon-Nikodym derivative. If μ 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 μ an arbitrary infinite cardinal number.

Mathematical Subject Classification 2000
Primary: 28A60
Secondary: 06E10, 28A12
Received: 9 December 1980
Revised: 20 May 1981
Published: 1 April 1982
Thomas E. Armstrong
Karel Libor Prikry