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

Louis Harvey Blake

Vol. 41 (1972), No. 1, 25–31

DOI: 10.2140/pjm.1972.41.25

Abstract

Let (Ω,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.P_e will be called a simple extension of P to A′ if P_e is a probability measure on A′ which 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 P_c to examine under what conditions the regularity of the conditional probability P^B will extend to the regularity of P_c^B. Also, if A is countably generated and P_c^B is regular, a characterization of P_c^B in terms of P^B will be given.

Mathematical Subject Classification 2000

Primary: 60A10

Secondary: 28A10

Milestones

Received: 30 November 1970

Published: 1 April 1972