  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.
