 
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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