A Consistency Theorem for Regular Conditional Distributions

Let (omega, beta) be a measurable space, An in B a sub-sigma-field and µn a random probability measure, n >= 1. In various frameworks, one looks for a probability P on B such that µn is a regular conditional distribution for P given An for all n. Conditions for such a P to exist are given. The conditions are quite simple when (omega, beta) is a compact Hausdorff space equipped with the Borel or the Bairesigma-field (as well as under other similar assumptions). Such conditions are then applied to Bayesian statistics.