On the equivalence of conglomerability and disintegrability for unbounded random variables

We extend a result of Dubins (Ann Probab 3:89–99, 1975) 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 randomvariables that have finite expected value andwhose conditional expectations are finite and have finite expected value.