Vol. 41, No. 1, 1972

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Canonical extensions of measures and the extension of regularity of conditional probabilities

Louis Harvey Blake

Vol. 41 (1972), No. 1, 25–31

Let ,A,P) be a probability space with B a sub σ-field of A. Let A′≡ σ(A,H), the a-field generated by A and H, where H is a subset of Ω not in A.Pe will be called a simple extension of P to Aif Pe is a probability measure on Awhich agrees with P on A.

The purpose of this paper is to use a particular type of simple extension called a canonical extension, denoted as Pc to examine under what conditions the regularity of the conditional probability PB will extend to the regularity of PcB. Also, if A is countably generated and PcB is regular, a characterization of PcB in terms of PB will be given.

Mathematical Subject Classification 2000
Primary: 60A10
Secondary: 28A10
Received: 30 November 1970
Published: 1 April 1972
Louis Harvey Blake