On the equivalence of conglomerability and disintegrability for unbounded random variables

<Para ID="Par1">We extend a result of Dubins (Ann Probab 3:89–99, <CitationRef CitationID="CR8">1975</CitationRef>) 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 collections of random variables that have finite expected value and whose conditional expectations are finite and have finite expected value. Copyright Springer-Verlag Berlin Heidelberg 2014