Let $F$ be a continuous cdf with full support on $[0,1].$ Let $A$ be a compact subset of $\mathbb{R}$ and $\mathcal{M}$ be the set of measurable functions $\alpha:[0,1]\rightarrow A.$ Let $\bar \alpha = \int_0^1 \alpha(x)dF(x).$ I am trying to prove that for $g:\mathbb{R}^3\rightarrow \mathbb{R}$ bounded and Lipschitz continuous,
$\max\limits_{\alpha\in \mathcal{M}} \int_0^1 g(\alpha(x),\bar \alpha, x)dF(x)$
has a solution. Would endowing the set $\mathcal{M}$ with the $L^1$ weak topology make it weakly compact? If so, do I need additional conditions on $g$ to guarantee existence of a maximizer?
