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

Louis Harvey Blake

Vol. 46 (1973), No. 2, 355–359

DOI: 10.2140/pjm.1973.46.355

Abstract

Throughout this paper, the following notation will be adopted. (Ω,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 A′ will be the σ-field generated by A and H. P_e will be a simple extension of P to A^r if P_e is a probability measure on A′ with P_e|_A = P.

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

(a) If P_e^B is regular for some canonical extension P₀ of P to A′, then P_e^B is regular for any simple extension P_e of P to A^f.

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

Mathematical Subject Classification 2000

Primary: 60A10

Milestones

Received: 22 February 1972

Published: 1 June 1973