Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T05:16:16.903Z Has data issue: false hasContentIssue false
  • English
  • Français

NONCONGLOMERABILITY FOR COUNTABLY ADDITIVE MEASURES THAT ARE NOT κ-ADDITIVE

Published online by Cambridge University Press:  27 December 2016

MARK J. SCHERVISH ,
TEDDY SEIDENFELD  and
JOSEPH B. KADANE
MARK J. SCHERVISH*
Affiliation:
Statistics Department, Carnegie Mellon University
TEDDY SEIDENFELD*
Affiliation:
Philosophy & Statistics Departments, Carnegie Mellon University
JOSEPH B. KADANE*
Affiliation:
Statistics Department, Carnegie Mellon University
*
*STATISTICS DEPARTMENT CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA 15213, USA E-mail: [email protected]
†PHILOSOPHY & STATISTICS DEPARTMENTS CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA 15213, USA E-mail: [email protected]
‡STATISTICS DEPARTMENT CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA 15213, USA E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let κ be an uncountable cardinal. Using the theory of conditional probability associated with de Finetti (1974) and Dubins (1975), subject to several structural assumptions for creating sufficiently many measurable sets, and assuming that κ is not a weakly inaccessible cardinal, we show that each probability that is not κ-additive has conditional probabilities that fail to be conglomerable in a partition of cardinality no greater than κ. This generalizes a result of Schervish, Seidenfeld, & Kadane (1984), which established that each finite but not countably additive probability has conditional probabilities that fail to be conglomerable in some countable partition.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 2 , June 2017 , pp. 284 - 300
Copyright
Copyright © Association for Symbolic Logic 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Armstrong, T. E. & Prikry, K. (1980). κ-finiteness and κ-additivity of measures on sets and left invariant measures on discrete groups. Proceedings of the American Mathematical Society, 80, 105–112.Google Scholar
Billingsley, P. (1995). Probability and Measure (third edition). New York: Wiley.Google Scholar
de Fineitti, B. (1974). Theory of Probability. New York: Wiley.Google Scholar
Doob, J. L. (1994). Measure Theory. New York: Springer-Verlag.Google Scholar
Dubins, L. (1975). Finitely additive conditional probabilities, conglomerability and disintegrations. Annals of Probability, 3, 89–99.Google Scholar
Fodor, G. (1956). Eine Bemerkung zur Theorie der regressiven Funktionen. Acta Scientiarum Mathematicarum (Szeged), 17, 139–142.Google Scholar
Halmos, P. R. (1950). Measure Theory. New York: Springer-Verlag.Google Scholar
Jech, T. (1978). Set Theory. New York: Academic Press.Google Scholar
Kadane, J. B., Schervish, M. J., & Seidenfeld, T. (1986). Statistical implications of finite additivity. In Goel, P. K. and Zellner, A., editors. Bayesian Inference and Decision Techniques. Amsterdam: Elsevier, pp. 59–76.Google Scholar
Kadane, J. B., Schervish, M. J., & Seidenfeld, T. (1996). Reasoning to a foregone conclusion. Journal of American Statistical Association, 91, 1228–1235.Google Scholar
Kolmogorov, A. (1956). Foundations of the Theory of Probability. New York: Chelsea.Google Scholar
Schervish, M. J., Seidenfeld, T., & Kadane, J. B. (1984). The extent of non-conglomerability of finitely additive probabilities. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 66(2), 205–226.Google Scholar
Seidenfeld, T., Schervish, M. J., & Kadane, J. B. (2001). Improper regular conditional distributions. Annals of Probability, 29, 1612–1624.Google Scholar
Ulam, S. (1930). Zur Masstheorie in der allgemeinen Mengenlehre. Fundamenta Mathematicae, 16, 140–150.CrossRefGoogle Scholar