Existence of compatible families of proper regular conditional probabilities

Let (Ω, ℱ, μ) be a perfect probability space with ℱ countably generated, and let IB be a family of sub-σ-fields of ℱ. Under a countability condition on the family IB, I show that there exists a family {π_∇}_∇∈IB of regular conditional probabilities which are everywhere compatible. Under a more stringent condition on IB, I show that the π_∇ can furthermore be chosen to be everywhere proper. It follows that in the Dobrushin-Lanford-Ruelle formulation of the statistical mechanics of classical lattice systems, every (perfect) probability measure is a Gibbs measure for some specification.