Improper regular conditional distributions. (English) Zbl 1017.60007

Let \((\Omega,\mathcal B,P)\) be a probability space and let \(\mathcal A\) be a conditioning sub-\(\sigma\)-field. An rcd (regular conditional distribution) \(P(.|\mathcal A)\) is said to be proper at \(\omega\in {\Omega}\) if \(P(A|{\mathcal A})(\omega)\) = 1 whenever \(\omega\in A\in {\mathcal A}\). If it is not everywhere proper, then it is said to be improper and if \(P(A|{\mathcal A})(\omega)\) = 0 for some \(\omega\in A\in {\mathcal A}\), then it is said to be maximally improper. In this rather technical paper, the authors explore the extent of local and global impropriety. Particular results deal with countably generated resp. symmetric sub-\(\sigma\)-fields. The concluding remarks are interesting.


60A10 Probabilistic measure theory
Full Text: DOI


[1] Billingsley, P. (1995). Probability and Measure, 3rd. ed. Wiley, New York. · Zbl 0822.60002
[2] Blackwell, D. (1955). On a class of probability spaces. Proc.Third Berkeley Symp.Math.Statist. Probab. 1-6. Univ. California Press, Berkeley. · Zbl 0073.12301
[3] Blackwell, D. and Dubins, L. E. (1975). On existence and non-existence of proper, regular, conditional distributions. Ann.Probab.3 741-752. · Zbl 0348.60003
[4] Blackwell, D. and Ryll-Nardzewski, C. (1963). Non-existence of everywhere proper conditional distributions. Ann.Math.Statist.34 223-225. · Zbl 0122.13202
[5] Breiman, L. (1968). Probability. Addison-Wesley, Reading, MA. · Zbl 0174.48801
[6] deFinetti, B. (1974). The Theory of Probability. Wiley, New York.
[7] Doob, J. L. (1953). Stochastic Processes. Wiley, New York. · Zbl 0053.26802
[8] Dubins, L. E. (1971). On conditional distributions for stochastic processes. In Proceedings of the Symposium in Probability Theory 72-85. Univ. Aarhus, Denmark. · Zbl 0245.60039
[9] Dubins, L. E. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. Ann.Probab.3 89-99. · Zbl 0302.60002
[10] Dubins, L. E. (1977). Measurable, tail disintegrations of the Haar integral are purely finite additive. Proc.Amer.Math.Soc.62 34-36. JSTOR: · Zbl 0349.28009
[11] Halmos, P. R. (1950). Measure Theory. van Nostrand, New York. · Zbl 0040.16802
[12] Hewitt, E. and Savage, L. J. (1955). Symmetric measures on cartesian products. Trans.Amer. Math.Soc.80 470-501. JSTOR: · Zbl 0066.29604
[13] Kadane, J. B., Schervish, M. J. and Seidenfeld, T. (1986). Statistical implications of finitely additive probability. In Bayesian Inference and Decision Techniques With Applications (P. Goel and A. Zellner, eds.) 59-76. North-Holland, Amsterdam. · Zbl 0619.62007
[14] Kolmogorov, A. (1950). Foundations of the Theory of Probability. Chelsea, New York. · Zbl 0074.12202
[15] Loeve, M. (1955). Probability Theory. van Nostrand, New York. · Zbl 0066.10903
[16] Savage, L. J. (1954). The Foundations of Statistics. Wiley, New York. · Zbl 0055.12604
[17] Schervish, M. J. (1995). Theory of Statistics. Springer, New York. · Zbl 0834.62002
[18] Schervish, M. J., Seidenfeld, T. and Kadane, J. B. (1984). The extent of non-conglomerability of finitely additive probabilities.Warsch.Verw.Gebiete 66 205-226. · Zbl 0525.60003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.