A Robust Version of Convex Integral Functionals

TL;DR

This paper develops a robust framework for convex integral functionals by providing bounds for their conjugates, analyzing regularity conditions, and extending classical results to include singular measures.

Contribution

It introduces bounds for the conjugate of convex integral functionals involving singular measures and explores conditions for their elimination, extending classical convex analysis results.

Findings

Bounds for conjugates include regular and singular parts.

Conditions identified when singular measures are eliminated.

Extensions of Rockafellar's classical results to broader settings.

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