The extent of nonconglomerability of finitely additive probabilities
 Mark J. Schervish,
 Teddy Seidenfeld,
 Joseph B. Kadane
 … show all 3 hide
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Summary
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 σfield, the bound is sharp. Hence, nonconglomerability (or equivalently nondisintegrability) 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 nonconglomerability 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.
 Armstrong, T.E., Prikry, K.: kFiniteness and kAdditivity of Measures on Sets and Left Invariant Measures on Discrete Groups. Proc. Amer. Math. Soc. 80, 105–112 (1980)
 Armstrong, T.E., Sudderth, W.: Nearly Strategic Measures. Pacific J of Math. Forthcoming (University of Minnesota School of Statistics Technical Report No. 334), (1981)
 Ash, R.B.: Real Analysis and Probability. New York: Academic Press 1972
 Bochner, S., Phillips, R.S.: Additive Set Functions and Vector Lattices. Ann. Math. 42, 316–324 (1941) CrossRef
 Comfort, W.W., Negrepontis, S.: The Theory of Ultrafilters. BerlinHeidelbergNew York: Springer 1974 CrossRef
 DeFinetti, B.: Probability, Induction and Statistics. New York: Wiley 1972
 DeFinetti, B.: The Theory of Probability. (2 volumes) New York: Wiley 1974
 Dubins, L.E.: Finitely Additive Conditional Probabilities, Conglomerability and Disintegrations. The Ann. Probability 3, 89–99 (1975) CrossRef
 Dunford, N., Schwartz, J.T.: Linear Operations, Part I: General Theory. New York: Interscience 1958
 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)
 Kolmogorov, A.N.: Foundations of the Theory of Probability. New York: Chelsea 1956
 Lindley, D.: Bayesian Statistics, A Review. Philadelphia: SIAM, 1971
 Lindley, D.: Comment on ‘Strong Inconsistency from Uniform Priors’. J. Amer. Statist. Assoc. 71, 120–121 (1976)
 Prikry, K., Sudderth, W.: Singularity with Respect to Strategic Measures. Unpublished manuscript (1980)
 Sobczyk, A., Hammer, P.: A Decomposition of Additive Set Functions. Duke J. Math. 11, 839–846 (1944) CrossRef
 Yosida, K., Hewitt, E.: Finitely Additive Measures. Trans. Amer. Math. Soc. 72, 46–66 (1952) CrossRef
 Title
 The extent of nonconglomerability of finitely additive probabilities
 Journal

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
Volume 66, Issue 2 , pp 205226
 Cover Date
 19840701
 DOI
 10.1007/BF00531529
 Print ISSN
 00443719
 Online ISSN
 14322064
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Mark J. Schervish ^{(1)}
 Teddy Seidenfeld ^{(1)}
 Joseph B. Kadane ^{(1)}
 Author Affiliations

 1. Statistics Department, CarnegieMellon University, 15213, Pittsburgh, PA, USA