Consider a function $f: \mathbb{R} \times \Delta(Q) \to \mathbb{R}$ and the following maximization problem:
$\max_{Q \in \Delta(Q)} \int f(x,Q^{(P)}) dP(x)$,
where P is a probability measure over the set $\Delta(P)$ and the notation $Q^{(P)}$ implies that $Q$ depends on the measure $P$. Which are the conditions on $f$ so the following holds
$\max_{Q \in \Delta(Q)} \int f(x,Q^{(P)}) dP(x) = \int \max_{Q \in \Delta(Q)}f(x,Q^{(P)}) dP(x)$.