The extent of non-conglomerability of finitely additive probabilities

  • Mark J. Schervish
  • Teddy Seidenfeld
  • Joseph B. Kadane


An arbitrary finitely additive probability can be decomposed uniquely into a convex combination of a countably additive probability and a purely finitely additive (PFA) one. The coefficient of the PFA probability is an upper bound on the extent to which conglomerability may fail in a finitely additive probability with that decomposition. If the probability is defined on a σ-field, the bound is sharp. Hence, non-conglomerability (or equivalently non-disintegrability) characterizes finitely as opposed to countably additive probability. Nonetheless, there exists a PFA probability which is simultaneously conglomerable over an arbitrary finite set of partitions.

Neither conglomerability nor non-conglomerability in a given partition is closed under convex combinations. But the convex combination of PFA ultrafilter probabilities, each of which cannot be made conglomerable in a common margin, is singular with respect to any finitely additive probability that is conglomerable in that margin.


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  1. 1.
    Armstrong, T.E., Prikry, K.: k-Finiteness and k-Additivity of Measures on Sets and Left Invariant Measures on Discrete Groups. Proc. Amer. Math. Soc. 80, 105–112 (1980)MathSciNetMATHGoogle Scholar
  2. 2.
    Armstrong, T.E., Sudderth, W.: Nearly Strategic Measures. Pacific J of Math. Forthcoming (University of Minnesota School of Statistics Technical Report No. 334), (1981)Google Scholar
  3. 3.
    Ash, R.B.: Real Analysis and Probability. New York: Academic Press 1972Google Scholar
  4. 4.
    Bochner, S., Phillips, R.S.: Additive Set Functions and Vector Lattices. Ann. Math. 42, 316–324 (1941)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. Berlin-Heidelberg-New York: Springer 1974CrossRefMATHGoogle Scholar
  6. 6.
    DeFinetti, B.: Probability, Induction and Statistics. New York: Wiley 1972Google Scholar
  7. 7.
    DeFinetti, B.: The Theory of Probability. (2 volumes) New York: Wiley 1974Google Scholar
  8. 8.
    Dubins, L.E.: Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations. The Ann. Probability 3, 89–99 (1975)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Dunford, N., Schwartz, J.T.: Linear Operations, Part I: General Theory. New York: Interscience 1958Google Scholar
  10. 10.
    Kadane, J.B., Schervish, M.J., Seidenfeld, T.: Statistical implications of finitely additive probability. In: Bayesian Inference and Decision Techniques with Applications: Essays in Honor of Bruno de Finetti, Goel, P.K., Zellner, A. (eds.) (to appear)Google Scholar
  11. 11.
    Kolmogorov, A.N.: Foundations of the Theory of Probability. New York: Chelsea 1956MATHGoogle Scholar
  12. 12.
    Lindley, D.: Bayesian Statistics, A Review. Philadelphia: SIAM, 1971MATHGoogle Scholar
  13. 13.
    Lindley, D.: Comment on ‘Strong Inconsistency from Uniform Priors’. J. Amer. Statist. Assoc. 71, 120–121 (1976)MathSciNetGoogle Scholar
  14. 14.
    Prikry, K., Sudderth, W.: Singularity with Respect to Strategic Measures. Unpublished manuscript (1980)Google Scholar
  15. 15.
    Sobczyk, A., Hammer, P.: A Decomposition of Additive Set Functions. Duke J. Math. 11, 839–846 (1944)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Yosida, K., Hewitt, E.: Finitely Additive Measures. Trans. Amer. Math. Soc. 72, 46–66 (1952)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Mark J. Schervish
    • 1
  • Teddy Seidenfeld
    • 1
  • Joseph B. Kadane
    • 1
  1. 1.Statistics DepartmentCarnegie-Mellon UniversityPittsburghUSA

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