Summary
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Doob, J.L.: Stochastic Processes, p. 26. New York: Wiley 1953
Hoffman-Jørgensen, J.: Existence of conditional probabilities. Math. Scand. 28, 257–264 (1971)
Chatterji, S.D.: Les martingales et leurs applications analytiques. In: Ecole d'été de Probabilités, Processus Stochastiques, Lecture Notes in Mathematics #307, p. 118. Berlin-Heidelberg-New York: Springer 1973
Kulakova, V.G.: The regularity of conditional probabilities [in Russian]. Vestnik Leningrad Univ. Mat. Mekh. Astronom., no. 1, vyp. 1, pp. 16–20 (1976). [English review in Math. Reviews 54, #14029]
Dobrushin, R.L.: The description of a random field by means of conditional probabilities and conditions of its regularity. Teor. Veroya. Prim. 13, 201–229 (1968) [Theor. Prob. Appl. 13, 197–224 (1968)]
Dobrushin, R.L.: Gibbsian random fields for lattice systems with pairwise interactions. Funkt. Anal. Prilozhen. 2, no. 4, 31–43 (1968) [Funct. Anal. Appl. 2, 292–301 (1968)]
Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Teor. Veroya. Prim. 15, 469–497 (1970) [Theor. Prob. Appl. 15, 458–486 (1970)]
Lanford, O.E. III, Ruelle, D.: Observables at infinity and states with short range correlations in statistical mechanics. Commun. Math. Phys. 13, 194–215 (1969)
Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)
Preston, C.: Random Fields. Lecture Notes in Mathematics #534. Berlin-Heidelberg-New York: Springer 1976
Föllmer, H.: Phase transition and Martin boundary. In: Séminaire de Probabilités IX, Université de Strasbourg, Lecture Notes in Mathematics #465, pp. 305–317. Berlin-Heidelberg-New York: Springer 1975
Goldstein, S.: A note on specifications. Z. Wahrscheinlichkeitstheorie verw. Geb. 46, 45–51 (1978)
Preston, C.: Construction of specifications. In: Quantum Fields — Algebras, Processes (Bielefeld symposium 1978), pp. 269–292, ed. L. Streit. Wien-New York: Springer 1980
Jiřina, M.: Conditional probabilities on σ-algebras with countable basis. Czech. Math. J. 4 (79), 372–380 (1954) [Selected Transitions in Mathematical Statistics and Probability, vol. 2 (Providence: American Mathematical Society, 1962), pp. 79–86]
Sazonov, V.V.: On perfect measures. Izv. Akad. Nauk SSSR Ser. Mat. 26, 391–414 (1962) [Amer. Math. Soc. Translations, series 2, no. 48, pp. 229–254 (1965)]
Darst, R.B.: On universal measurability and perfect probability. Ann. Math. Statist. 42, 352–354 (1971)
Blackwell, D.: On a class of probability spaces. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 1–6, ed. Jerzy Neyman. Berkeley and Los Angeles: University of California Press 1956
Parthasarathy, K.R.: Probability Measures on Metric Spaces, chapter 5. New York: Academic Press 1967
Halmos, P.R.: The decomposition of measures. Duke Math. J. 8, 386–392 (1941)
Blackwell, D., Ryll-Nardzewski, C.: Non-existence of everywhere proper conditional distributions. Ann. Math. Statist. 34, 223–225 (1963)
Blackwell, D., Dubins, L.E.: On existence and non-existence of proper, regular, conditional distributions. Ann. Probability 3, 741–752 (1975)
Ramachandran, D.: Existence of independent complements in regular conditional probaiblity spaces. Ann. Probability 7, 433–443 (1979)
Neveu, J.: Bases Mathématiques du Calcul des Probabilités, Proposition III-2-1 and Corollaire 2. Paris: Masson 1964, 2eme ed. 1970 [Mathematical Foundations of the Calculus of Probability. San Francisco: Holden-Day 1965]
Bourbaki, N.: Elements of Mathematics, Theory of Sets (pp. 190, 249–250). Section III.6.5 and Exercise III.6.29. Paris: Hermann and Addison-Wesley 1968
Kuznetsov, S.E.: Any Markov process in a Borel space has a transition function. Teor. Veroya. Prim. 25, 389–393 (1980) [Theor. Prob. Appl. 25, no. 2 (to appear)]
Author information
Authors and Affiliations
Additional information
Research supported in part by NSF PHY-78-23952
NSF Predoctoral Fellow (1976–79) and Danforth Fellow (1979–81).
Rights and permissions
About this article
Cite this article
Sokal, A.D. Existence of compatible families of proper regular conditional probabilities. Z. Wahrscheinlichkeitstheorie verw Gebiete 56, 537–548 (1981). https://doi.org/10.1007/BF00531432
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00531432