Let $X\subseteq\mathbb R^d$ be a compact and $Y=\mathbb R^d.$ Let $\Gamma:X\twoheadrightarrow Y$ be a multi-valued map with closed values. Assume that $\Gamma$ admits a continuous (single-valued) selector. Can we conclude that $\Gamma$ is measurable?
I call this the reverse-reverse of Michael selection theorem since the reverse of this theorem says that if $\Gamma$ is continuous and admits a continuous selector, then $X$ must be compact. So what I want is a sort of reverse-reverse of this last statement...