Abstract： 

We consider the pointwise supremum of a family of convex integral functionals of essentially bounded random variables, each associated to a common convex integrand and a respective probability measure belonging to a dominated weakly compact convex set. Its conjugate functional is analyzed, providing a pair of upper and lower bounds as direct sums of common regular part and respective singular parts, which coincide when the defining set of probabilities is a singleton, as the classical Rockafellar theorem, and these bounds are generally the best in a certain sense. We then investigate when the conjugate eliminates the singular measures, which a fortiori implies the equality of the upper and lower bounds, and its relation to other finer regularity properties of the original functional and of the conjugate. As an application, a general duality result in the robust utility maximization problem is obtained. 