Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

, Volume 66, Issue 2, pp 205–226

The extent of non-conglomerability of finitely additive probabilities

Authors

  • Mark J. Schervish
    • Statistics DepartmentCarnegie-Mellon University
  • Teddy Seidenfeld
    • Statistics DepartmentCarnegie-Mellon University
  • Joseph B. Kadane
    • Statistics DepartmentCarnegie-Mellon University
Article

DOI: 10.1007/BF00531529

Cite this article as:
Schervish, M.J., Seidenfeld, T. & Kadane, J.B. Z. Wahrscheinlichkeitstheorie verw Gebiete (1984) 66: 205. doi:10.1007/BF00531529

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 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.

Copyright information

© Springer-Verlag 1984