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Summary: March 2, 2009
ON THE EQUIVALENCE OF CONGLOMERABILITY AND
DISINTEGRABILITY FOR UNBOUNDED RANDOM
VARIABLES.
BY MARK J. SCHERVISH, TEDDY SEIDENFELD AND JOSEPH B. KADANE
Carnegie Mellon University
We extend a result of Dubins (1975) from bounded to unbounded random
variables. Dubins (1975) showed that a finitely additive expectation over
the collection of bounded random variables can be written as an integral
of conditional expectations (disintegrability) if and only if the marginal ex-
pectation is always within the smallest closed interval containing the con-
ditional expectations (conglomerability). We give a sufficient condition to
extend this result to the collection Z of all random variables that have fi-
nite expected value and whose conditional expectations are finite and have
SHORT TITLE: CONGLOMERABILITY AND DISINTEGRABILITY
MSC 2000 subject classifications. Primary 60A05; secondary 28C05.
Key words and phrases. finite additivity, law of total probability, Daniell integral, coherence,
ultrafilters.
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