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)
I
f
,
γ
(
ξ
)
=
sup
Q
(
∫
Ω
f
(
ω
,
ξ
(
ω
)
)
Q
(
d
ω
)
−
γ
(
Q
)
)
on
L^\infty(\Omega,\mathcal{F},\mathbb{P})
L
∞
(
Ω
,
F
,
P
)
where
f:\Omega \times\mathbb{R}\rightarrow\overline{\mathbb{R}}
f
:
Ω
×
R
→
R
‾
is a proper normal convex integrand,
\gamma
γ
is a proper convex function on the set of probability measures absolutely continuous w.r.t.
\mathbb{P}
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}
I
f
,
γ
as direct sums of a common regular part and respective singular parts; they coincide when
\mathrm{dom}(\gamma)=\{\mathbb{P}\}
d
o
m
(
γ
)
=
{
P
}
as Rockafellar's classical 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.