Existence of Borel measurable argmax-function given semicontinuity Let $\mathcal{S}$ and $\mathcal{A}$ be standard Borel (that is, a Polish space endowed with the sigma algebra generated by the open sets its metric topology), and $\mathcal{A}$ be compact.
Suppose $f : \mathcal{S} \times \mathcal{A} \to \mathbb{R}$ is upper semicontinuous.
Then the set $A_s = \mathrm{argmax}_{a \in \mathcal{A}} f(s, a) \neq \emptyset$ is non-empty for any $s \in \mathcal{S}$.
Q: Does there exist a measurable function $\phi : \mathcal{S} \to \mathcal{A}$ such that
$\phi(s) \in A_s$ for all $s \in \mathcal{S}$ ?
By axiom of choice one may pick at least one function $\psi : \mathcal{S} \to \mathcal{A}$ with $\psi(s) \in A_s, \;\forall s \in \mathcal{S}$.
However I am lost as to how to know if any such function can be chosen to be measurable.

The answer is yes and is the question is basically proposition 7.33 in
this book.
Thanks to @Giuseppe Negro, for pointing me to the source.
