Existence of compatible families of proper regular conditional probabilities

Alan D. Sokal

    Research output: Contribution to journalArticle

    Abstract

    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
    Volume56
    Issue number4
    DOIs
    StatePublished - Dec 1981

    Fingerprint

    Conditional probability
    Gibbs Measure
    Lattice System
    Subfield
    Probability Space
    Statistical Mechanics
    Probability Measure
    Specification
    Formulation
    Family

    ASJC Scopus subject areas

    • Statistics and Probability
    • Analysis
    • Mathematics(all)

    Cite this

    Existence of compatible families of proper regular conditional probabilities. / Sokal, Alan D.

    In: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 56, No. 4, 12.1981, p. 537-548.

    Research output: Contribution to journalArticle

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