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 a 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.
