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Countably additive full conditional probabilities

Author: Thomas E. Armstrong
Journal: Proc. Amer. Math. Soc. 107 (1989), 977-987
MSC: Primary 60A10
MathSciNet review: 975631
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Abstract: Armstrong-Sudderth showed that a method of Carlson could be adapted to construct examples of countably additive full conditional probabilities. Here it is shown that all countably additive full conditional probabilities arise in this fashion. It is also shown that if the dimensionally ordered family of measures associated by Renyi to a full conditional probability contains an unbounded measure the full conditional probability fails to be countably additive. This leads to the notion of proper full conditional probabilities which if well founded in the sense of Krauss agree with the class of full conditional probabilities arising from Carlson's construction. Section 3 explores the connection between coherence of countable betting systems and countable additivity of full conditional probabilities.

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  • [1] T. E. Armstrong, When do the regular sets for a finitely additive Borel measure form a $ \sigma $algebra?, J. Austral. Math. Soc. Ser. A 33 (1982), 374-385. MR 678515 (84f:28010)
  • [2] T. E. Armstrong and K. Prikry, $ \kappa $-additivity and $ \kappa $-finiteness of measures on sets and left invariant measures on discrete groups, Proc. Amer. Math. Soc. 80 (1980), 105-112. MR 574517 (81k:28014)
  • [3] T. E. Armstrong and W. Sudderth, Locally coherent rates of exchange, Annals of Statistics 17, (1989) (to appear). MR 1015160 (90k:62009)
  • [4] T. Carlson, A solution of Ulam's problem on relative measure, Proc. Amer. Math. Soc. 94 (1985), 129-134. MR 781070 (86m:03080)
  • [5] B. deFinetti, Probability, induction, and statistics, Wiley, New York, 1972.
  • [6] L. E. Dubins, Finitely additive conditional probabilities, conglomerability and disintegrations, Ann. Probab. 3 (1975), 89-99. MR 0358891 (50:11348)
  • [7] H. J. Falconer, The geometry of fractal sets, Cambridge University Press, Cambridge, 1985.
  • [8] J. Hartigan, Bayes theory, Springer-Verlag, Berlin-New York, 1983. MR 715782 (85k:60008)
  • [9] D. Heath and W. Sudderth, On finitely additive priors, coherence, and extended admissibility, Ann. Statist. 6 (1978), 333-345. MR 0464450 (57:4380)
  • [10] P. H. Krauss, Representation of conditional probability measures on Boolean algebras, Acta Math. Hungar. 19 (1968), 229-241. MR 0236080 (38:4378)
  • [11] B. B. Mandelbrot, Fractals: form, chance and dimension, Freeman, San Francisco, 1977. MR 0471493 (57:11224)
  • [12] A. Renyi, On a new axiomatic theory of probability, Acta Math. Hungar. 6 (1955), 285-355. MR 0081008 (18:339h)
  • [13] -, On conditional probability spaces generated by a dimensionally ordered set of measures, Theory Probab. Appl. 1 (1956), 61-71. MR 0085639 (19:69j)
  • [14] -, Foundations of probability, Holden-Day, San Francisco, 1970. MR 0264723 (41:9314)
  • [15] C. A. Rogers, Hausdorff measures, Cambridge University Press, Cambridge, 1970. MR 0281862 (43:7576)

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Keywords: Full conditional probability, countable additivity, coherence
Article copyright: © Copyright 1989 American Mathematical Society

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