A consistency theorem for regular conditional distributions

Let $(Omega,mathcal{B})$ be a measurable space, $mathcal{A}_nsubsetmathcal{B}$ a sub-$sigma$-field and $mu_n$ a random probability measure on $(Omega,mathcal{B})$, $ngeq 1$. In various frameworks, one looks for a probability $P$ on $mathcal{B}$ such that $mu_n$ is a regular conditional distribution for $P$ given $mathcal{A}_n$ for all $n$. Conditions for such a $P$ to exist are given. The conditions are quite simple when $(Omega,mathcal{B})$ is a compact Hausdorff space equipped with the Borel or the Baire $sigma$-field (as well as under similar assumptions). Applications to Gibbs measures and Bayesian statistics are given as well.