David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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We extend a result of Dubins  from bounded to unbounded random variables. Dubins  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 expectation is always within the smallest closed interval containing the conditional expectations (conglomerability). We give a suﬃcient condition to extend this result to the collection Z of all random variables that have ﬁnite expected value and whose conditional expectations are ﬁnite and have ﬁnite expected value. The suﬃcient condition also allows the result to extend some, but not all, subcollections of Z. We give an example where the equivalence of disintegrability and conglomerability fails for a subcollection of Z that still contains all bounded random variables.
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