Suppose that $\mathcal{C}$ and $\mathcal{D}$ are subsets of $L^2(X,\Sigma,\mu)\cap L^{\infty}(X,\Sigma,\mu)$, where $\mu$ is a finite-measure on $(X,\Sigma)$. Let $F:L^2(X,\Sigma,\mu)\times L^2(X,\Sigma,\mu)\rightarrow \mathbb{R}$ be of the form $$ F(f,g)\triangleq \int_{x \in X} G(f(x),g(x)) \mu(dx), $$ for some continuous map $G:\mathbb{R}^2 \rightarrow \mathbb{R}$.
Assume that:
- The map $y\mapsto \operatorname{argmin}_{f \in \mathcal{D}}F(g,f)$ is single-valued on $\mathcal{D}$ and (depends on the choice of $g$),
- The assumptions of Sion's Minimax Theorem hold.
Does this guarantee that $$ \operatorname{argmin}_{g \in \mathcal{C}}\operatorname{sup}_{f \in \mathcal{D}} F(g,f) = \operatorname{esssup}_{g \in \mathcal{D}} \operatorname{argmin}_{f \in \mathcal{C}}F(g,f) ? $$
If this is not true; then what additional assumptions are needed for it to hold?
