## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 249-264 |

Number of pages | 16 |

Journal | Pacific Journal of Mathematics |

Volume | 99 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1982 |