Countably additive full conditional probabilities
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- by Thomas E. Armstrong PDF
- Proc. Amer. Math. Soc. 107 (1989), 977-987 Request permission
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.References
- Thomas E. Armstrong, When is the algebra of regular sets for a finitely additive Borel measure a $\sigma$-algebra?, J. Austral. Math. Soc. Ser. A 33 (1982), no. 3, 374–385. MR 678515
- Thomas E. Armstrong and Karel Prikry, $\kappa$-finiteness and $\kappa$-additivity of measures on sets and left invariant measures on discrete groups, Proc. Amer. Math. Soc. 80 (1980), no. 1, 105–112. MR 574517, DOI 10.1090/S0002-9939-1980-0574517-0
- Thomas E. Armstrong and William D. Sudderth, Locally coherent rates of exchange, Ann. Statist. 17 (1989), no. 3, 1394–1408. MR 1015160, DOI 10.1214/aos/1176347278
- Tim Carlson, A solution of Ulam’s problem on relative measure, Proc. Amer. Math. Soc. 94 (1985), no. 1, 129–134. MR 781070, DOI 10.1090/S0002-9939-1985-0781070-X B. deFinetti, Probability, induction, and statistics, Wiley, New York, 1972.
- Lester E. Dubins, Finitely additive conditional probabilities, conglomerability and disintegrations, Ann. Probability 3 (1975), 89–99. MR 358891, DOI 10.1214/aop/1176996451 H. J. Falconer, The geometry of fractal sets, Cambridge University Press, Cambridge, 1985.
- J. A. Hartigan, Bayes theory, Springer Series in Statistics, Springer-Verlag, New York, 1983. MR 715782, DOI 10.1007/978-1-4613-8242-3
- David Heath and William Sudderth, On finitely additive priors, coherence, and extended admissibility, Ann. Statist. 6 (1978), no. 2, 333–345. MR 464450
- P. H. Krauss, Representation of conditional probability measures on Boolean algebras, Acta Math. Acad. Sci. Hungar. 19 (1968), 229–241. MR 236080, DOI 10.1007/BF01894506
- Benoit B. Mandelbrot, Fractals: form, chance, and dimension, Revised edition, W. H. Freeman and Co., San Francisco, Calif., 1977. Translated from the French. MR 0471493
- Alfréd Rényi, On a new axiomatic theory of probability, Acta Math. Acad. Sci. Hungar. 6 (1955), 285–335 (English, with Russian summary). MR 81008, DOI 10.1007/BF02024393
- A. Rényi, On conditional probability spaces generated by a dimensionally ordered set of measures, Teor. Veroyatnost. i Primenen. 1 (1956), 61–71 (English, with Russian summary). MR 0085639
- Alfréd Rényi, Foundations of probability, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1970. MR 0264723
- C. A. Rogers, Hausdorff measures, Cambridge University Press, London-New York, 1970. MR 0281862
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 107 (1989), 977-987
- MSC: Primary 60A10
- DOI: https://doi.org/10.1090/S0002-9939-1989-0975631-6
- MathSciNet review: 975631