We study the pointwise supremum of convex integral functionals $\mathcal{ I}_{ f,\gamma}(\xi)= \sup_{ Q} \left( \int_\Omega f(\omega,\xi(\omega))Q(d\omega)-\gamma(Q)\right)$ on $L^\infty(\Omega,\mathcal{ F},\mathbb{ P})$ where $f:\Omega\times\mathbb{ R}\rightarrow\overline{ \mathbb{ R}}$ is a proper normal convex integrand, $\gamma$ is a proper convex function on the set of probability measures absolutely continuous w.r.t. $\mathbb{ P}$, and the supremum is taken over all such measures. We give a pair of upper and lower bounds for the conjugate of $\mathcal{ I}_{ f,\gamma}$ as direct sums of a common regular part and respective singular parts; they coincide when $\mathrm{ dom}(\gamma)=\{ \mathbb{ P}\}$ as Rockafellar's result, while both inequalities can generally be strict. We then investigate when the conjugate eliminates the singular measures, which a fortiori yields the equality in bounds, and its relation to other finer regularity properties of the original functional and of the conjugate.
Categories: math.FA math.OC math.PR q-fin.CP