The Annals of Probability

0–1 laws for regular conditional distributions

Patrizia Berti and Pietro Rigo

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Let $(Ω, ℬ, P)$ be a probability space, $\mathscr{A}\subset\mathscr{B}$ a sub-$σ$-field, and $μ$ a regular conditional distribution for $P$ given $\mathscr{A}$. Necessary and sufficient conditions for $μ(ω)(A)$ to be 0–1, for all $A\in\mathscr{A}$ and $ω∈A_0$, where $A_{0}\in\mathscr{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $\mathscr{A}$ is a tail sub-$σ$-field. Let $H(ω)$ denote the $\mathscr{A}$-atom including the point $ω∈Ω$. Necessary and sufficient conditions for $μ(ω)(H(ω))$ to be 0–1, for all $ω∈A_0$, are also given. If $(Ω, ℬ)$ is a standard space, the latter 0–1 law is true for various classically interesting sub-$σ$-fields $\mathscr{A}$, including tail, symmetric, invariant, as well as some sub-$σ$-fields connected with continuous time processes.

Article information

Ann. Probab., Volume 35, Number 2 (2007), 649-662.

First available in Project Euclid: 30 March 2007

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Zentralblatt MATH identifier

Primary: 60A05: Axioms; other general questions 60A10: Probabilistic measure theory {For ergodic theory, see 28Dxx and 60Fxx} 60F20: Zero-one laws

0–1 law measurability regular conditional distribution tail σ-field


Berti, Patrizia; Rigo, Pietro. 0–1 laws for regular conditional distributions. Ann. Probab. 35 (2007), no. 2, 649--662. doi:10.1214/009117906000000845.

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  • Berti, P. and Rigo, P. (1999). Sufficient conditions for the existence of disintegrations. J. Theoret. Probab. 12 75–86.
  • Blackwell, D. (1956). On a class of probability spaces. In Proc. Third Berkeley Symp. Math. Statist. Probab. 1–6. Univ. California Press, Berkeley.
  • Blackwell, D. and Ryll-Nardzewski, C. (1963). Non-existence of everywhere proper conditional distributions. Ann. Math. Statist. 34 223–225.
  • Blackwell, D. and Dubins, L. E. (1975). On existence and non-existence of proper, regular, conditional distributions. Ann. Probab. 3 741–752.
  • Diaconis, P. and Freedman, D. (1984). Partial exchangeability and sufficiency. In Proc. of the Indian Statistical Institute Golden Jubilee International Conference on Statistics: Applications and New Directions 205–236. Indian Statist. Inst., Calcutta.
  • Dubins, L. E. and Freedman, D. (1964). Measurable sets of measures. Pacific J. Math. 14 1211–1222.
  • Dynkin, E. B. (1978). Sufficient statistics and extreme points. Ann. Probab. 6 705–730.
  • Fremlin, D. H. (1981). Measurable functions and almost continuous functions. Manuscripta Math. 33 387–405.
  • Maitra, A. (1977). Integral representations of invariant measures. Trans. Amer. Math. Soc. 229 209–225.
  • Sazonov, V. V. (1965). On perfect measures. Amer. Math. Soc. Transl. Ser. (2) 48 229–254.
  • Seidenfeld, T., Schervish, M. J. and Kadane, J. (2001). Improper regular conditional distributions. Ann. Probab. 29 1612–1624.