Canonical extensions of measures and the extension of regularity of conditional probabilities
Louis H. Blake · Pacific Journal of Mathematics · 1972
Let (Ω, % P) be a probability space with S3 a sub σ-field of 31.Let 91' = σ(% H), the σ-field generated by SI and H, where H is a subset of Ω not in %.P e will be called a simple extension of P to %' if P e is a probability measure on 51' which agrees with P on %.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 s will extend to the regularity of P c s .Also, if % is countably generated and P<? is regular, a characterization of P<? in terms of P 58 will be given.