sufficient and necessary condition to change the order of minimization (infimum)

Question

We want to know the sufficient and necessary condition of $\inf_{F\in set A}\inf_{0\leq p\leq 1}g(F,p)=\inf_{0\leq p\leq 1}\inf_{F\in set A}g(F,p)$, where $F$ is a function and set $A$ is a set of functionals. One potential answer is that set $A$ is compact and $g(F,p)$ is lower semi-continuous for both $F$ and $p$. But I don't know how to prove it. Or is there any references?