Existence of compatible families of proper regular conditional probabilities

Alan D. Sokal

    Research output: Contribution to journalArticlepeer-review


    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.

    Original languageEnglish (US)
    Pages (from-to)537-548
    Number of pages12
    JournalZeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
    Issue number4
    StatePublished - Dec 1981

    ASJC Scopus subject areas

    • Analysis
    • Statistics and Probability
    • General Mathematics


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