Regular conditional probability, disintegration of probability and Radon spaces
DOI:
https://doi.org/10.4067/S0716-09172004000100002Keywords:
Regular conditional probability, Radon space, separable Hausdorff measurable space, disintegration of probability.Abstract
We establish equivalence of several regular conditional probability properties and Radon space. In addition, we introduce the universally measurable disintegration concept and prove an existence result.References
[1] Bertsekas D. P., Shreve S. E. - Stochastic Optimal Control: The Discrete-Time Case, Academic Press, New York, (1978).
[2] D. Blackwell - On a class of probability spaces, Proc. Third Berkeley Symp. Math. Statist. Prob., pp. 1-6, University of California Press, (1955).
[3] D. Blackwell and C. Ryll-Nardzewski - Non-existence of everywhere proper conditional distribuition, Ann. Math. Statistics, Vol. 34, pp. 223-225, (1963).
[4] D. Blackwell and L. E. Dubins - On existence and non-existence of proper, regular, conditional distributions, Annals Probability, Vol. 3, No. 5, pp. 741-752, (1975).
[5] S. D. Chatterji - Disintegration of measures and lifting, Vector and Operator Valued Measures and Applications, Eds. D. H. Tucker and H. B. Naymard (Proc. Symp. Alta, Utah, 1972, pp. 69-83), Academic Press, New York, (1973).
[6] J. Dieudonné - Sur le théoreme de Lebesgue-Nikodym III, Ann. Univ. Grenoble, Vol. 23, pp. 25-53, (1948).
[7] J. L. Doob - Stochastic Processes, John Wiley ans Sons, INC., (1953).
[8] A. M. Faden - The existence of regular conditional probabilities: necessary and sufficient conditions , Annals Probability, Vol. 13, no. 1, pp. 288-298, (1985).
[9] S. Graf and G. M¨argel - Disintegration of a measure with respect to a correspondence, Lecture Notes in Math., Vol. 794, Springer-Verlag, pp. 167-169, (1980).
[10] P. K. Getoor - On the Construction of Kernels, Lecture Notes in Mathematics, Séminaire de Probabilités IX, Vol. 465, pp. 443-463, (1975).
[11] J. Hoffmann-J¨orgensen - Existence of Conditional Probabilities, Math. Scand., Vol. 28, pp. 257-264, (1971).
[12] M. Jirina - Probabilités conditionnelles sur des algebres a base dénombrable, Czechosloviak Math. J., Vol. 4, pp. 372-380, (1954).
[13] A. N. Kolmogorov - Grundbegreffi der Wahrscheinlichkeitsrechnung, Springer, Berlin, (1933).
[14] D. Leao - Radon spaces, Ph.D. thesis, State University of Campinas UNICAMP, Brazil, Nov. (1999).
[15] D. Leao, M. D. Fragoso and P. Ruffino - Characterization of Radon spaces, Statistics and Probability Letters, Vol. 42, pp. 409-413, (1999).
[16] S. J. Leese - Measurable selections and the uniformization of Souslin sets, American Journal of Mathematics, Vol. 1000, No. 1, pp. 19-41, (1978).
[17] K. Musial - Existence of proper conditional probabilities, Z. Wahrscheinlichkeitstheorie verw. Geb., Vol. 22, pp. 8-12, (1972)
[18] J. von Neumann - Zur operatorenmethode in der Klassischen Mechanik, Annals of Math., vol. 33, pp. 587-642, (1932).
[19] J. K. Pachl - Disintegration and compact measures, Math. Scand., Vol. 43, pp. 157-168, (1978).
[20] V. V. Sazonov - On perfect measures, Amer. Math. Soc. Transl. Series 2, Vol. 48, pp. 229-254, (1965).
[21] M. M. Valadier - Désintegration d’une mesure sur un produit, C. R. Acad. Sc. Paris, série A, vol. 276, pp. 33-35, (1973).
[2] D. Blackwell - On a class of probability spaces, Proc. Third Berkeley Symp. Math. Statist. Prob., pp. 1-6, University of California Press, (1955).
[3] D. Blackwell and C. Ryll-Nardzewski - Non-existence of everywhere proper conditional distribuition, Ann. Math. Statistics, Vol. 34, pp. 223-225, (1963).
[4] D. Blackwell and L. E. Dubins - On existence and non-existence of proper, regular, conditional distributions, Annals Probability, Vol. 3, No. 5, pp. 741-752, (1975).
[5] S. D. Chatterji - Disintegration of measures and lifting, Vector and Operator Valued Measures and Applications, Eds. D. H. Tucker and H. B. Naymard (Proc. Symp. Alta, Utah, 1972, pp. 69-83), Academic Press, New York, (1973).
[6] J. Dieudonné - Sur le théoreme de Lebesgue-Nikodym III, Ann. Univ. Grenoble, Vol. 23, pp. 25-53, (1948).
[7] J. L. Doob - Stochastic Processes, John Wiley ans Sons, INC., (1953).
[8] A. M. Faden - The existence of regular conditional probabilities: necessary and sufficient conditions , Annals Probability, Vol. 13, no. 1, pp. 288-298, (1985).
[9] S. Graf and G. M¨argel - Disintegration of a measure with respect to a correspondence, Lecture Notes in Math., Vol. 794, Springer-Verlag, pp. 167-169, (1980).
[10] P. K. Getoor - On the Construction of Kernels, Lecture Notes in Mathematics, Séminaire de Probabilités IX, Vol. 465, pp. 443-463, (1975).
[11] J. Hoffmann-J¨orgensen - Existence of Conditional Probabilities, Math. Scand., Vol. 28, pp. 257-264, (1971).
[12] M. Jirina - Probabilités conditionnelles sur des algebres a base dénombrable, Czechosloviak Math. J., Vol. 4, pp. 372-380, (1954).
[13] A. N. Kolmogorov - Grundbegreffi der Wahrscheinlichkeitsrechnung, Springer, Berlin, (1933).
[14] D. Leao - Radon spaces, Ph.D. thesis, State University of Campinas UNICAMP, Brazil, Nov. (1999).
[15] D. Leao, M. D. Fragoso and P. Ruffino - Characterization of Radon spaces, Statistics and Probability Letters, Vol. 42, pp. 409-413, (1999).
[16] S. J. Leese - Measurable selections and the uniformization of Souslin sets, American Journal of Mathematics, Vol. 1000, No. 1, pp. 19-41, (1978).
[17] K. Musial - Existence of proper conditional probabilities, Z. Wahrscheinlichkeitstheorie verw. Geb., Vol. 22, pp. 8-12, (1972)
[18] J. von Neumann - Zur operatorenmethode in der Klassischen Mechanik, Annals of Math., vol. 33, pp. 587-642, (1932).
[19] J. K. Pachl - Disintegration and compact measures, Math. Scand., Vol. 43, pp. 157-168, (1978).
[20] V. V. Sazonov - On perfect measures, Amer. Math. Soc. Transl. Series 2, Vol. 48, pp. 229-254, (1965).
[21] M. M. Valadier - Désintegration d’une mesure sur un produit, C. R. Acad. Sc. Paris, série A, vol. 276, pp. 33-35, (1973).
Published
2017-05-22
How to Cite
[1]
D. Leao, Jr., M. D. Fragoso, and R. C. Ruffino, “Regular conditional probability, disintegration of probability and Radon spaces”, Proyecciones (Antofagasta, On line), vol. 23, no. 1, pp. 15-29, May 2017.
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