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

A finitely additive probability measure μ on a Boolean algebra B induces a semi-metric d_μ defined by d_μ(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 B_μ is the quotient of B via μ-null sets then B_μ is d_μ-complete iff μ is countably additive on B_μ and B_μ is complete as a Boolean algebra. Furthermore B_μ is d_μ-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 B_μ is not d_μ-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 B_μ an arbitrary infinite cardinal number.