Let $G$ be a Polish group, consider the standard Borel space $F(G)$ of closed subsets of $G$ whose borel structure is generated by the sets of the form $\{F\in F(G)\mid F\cap U \neq \emptyset\}$ for $U$ open in $G$.
It is stated in Gao's Invariant descriptive set theory (chapter 3.3) that the action $g\cdot F = Fg^{-1}$ of $G$ on $F(G)$ is Borel. I cannot see how this is so. It is clearly enough to show that for any open set $U$ we have that $\{(g, F)\mid (g\cdot F) \cap U \neq \emptyset\}$ is Borel, but I just cannot do it.
Thanks for any feedback.
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