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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: mark@stat.cmu.edu
†PHILOSOPHY & STATISTICS DEPARTMENTS CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA 15213, USA E-mail: teddy@stat.cmu.edu
‡STATISTICS DEPARTMENT CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA 15213, USA E-mail: kadane@stat.cmu.edu
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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 

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References

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