TY - JOUR

T1 - On the semimetric on a boolean algebra induced by a finitely additive probability measure

AU - Armstrong, Thomas E.

AU - Prikry, Karel

PY - 1982/4

Y1 - 1982/4

N2 - 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.

AB - 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.

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U2 - 10.2140/pjm.1982.99.249

DO - 10.2140/pjm.1982.99.249

M3 - Article

AN - SCOPUS:84972513598

VL - 99

SP - 249

EP - 264

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 2

ER -