On the equivalence of conglomerability and disintegrability for unbounded random variables
We extend a result of Dubins (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 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.