Let P be a convex set of finitely additive probability measures defined on a Boolean algebra of subsets of a set X of consequences. Axioms are specified for a preference relation $\prec$ on P which are necessary and sufficient for the existence of a real-valued utility function u on X for which expected utility E (u, p) is finite for all p in P and for which $p\precq$ iff E (u, p) > E (u, q), for all p and q in P. A slightly simpler set of axioms yields the same results when the algebra is a Borel algebra and every measure in P is countably additive. The axioms allow P to contain nonsimple probability measures without necessarily implying that the utility function u is bounded.
"Unbounded Expected Utility." Ann. Statist. 3 (4) 884 - 896, July, 1975. https://doi.org/10.1214/aos/1176343189