Definition 1
Let $F:X\to2^Y$ be a set-valued map from a metric space to the subsets of another metric space. We say it is upper semi-continuous (USC) if for every $\epsilon$ and every $x_0\in X$ there exists $\delta$ such that $d(x,x_0)<\delta\implies F(x)\subseteq B(F(x_0),\epsilon)=\bigcup_{t\in F(x_0)}B(t,\epsilon)$.
Definition 2
$F:X\to2^Y$ with the properties above (except semicontinuity) is weakly measurable (WM) if the preimage of an open set is Borel, that is, if $V$ open implies $\{x:F(x)\cap V\neq\varnothing\}$ is Borel.
Context
The Kuratowski-Ryll-Nardzewski theorem works for WM functions. I need to apply it to a USC function (which is also convex-compact-valued, if it may be of interest) to construct a selection for a set function originating in a control problem, as described here. So I would need the implication $USC\implies WM$.
Question
Is a USC correspondence necessarily WM?
