Vol. 46, No. 2, 1973

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Simple extensions of measures and the preservation of regularity of conditional probabilities

Louis Harvey Blake

Vol. 46 (1973), No. 2, 355–359

Throughout this paper, the following notation will be adopted. ,A,P) will be a probability space with B a sub σ-field of A.H will denote a subset of Ω not in A and Awill be the σ-field generated by A and H. Pe will be a simple extension of P to Ar if Pe is a probability measure on Awith Pe|A = P.

The ability to extend the regularity of the conditional probability PB to regularity of PeB has been explored earlier for canonical extensions of measures. The main results of this paper are:

(a) If PeB is regular for some canonical extension P0 of P to A, then PeB is regular for any simple extension Pe of P to Af.

(b) For some choice of ,A,P),B and H,PB is regular but for no Pe is PeB regular. This will essentially extend the Dicudonné example,

Mathematical Subject Classification 2000
Primary: 60A10
Received: 22 February 1972
Published: 1 June 1973
Louis Harvey Blake