A Robust Version of Convex Integral Functionals

Abstract

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 $f:\Omega \times \mathbb{R}\to \overline{\mathbb{R}}$ 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.

Minor typos are corrected. To appear in Journal of Convex Analysis