Let $(X,d)$ be a compact metric space, and let $\mathcal{K}(X)$ denote the set of compact subsets of $X$. We endow $\mathcal{K}(X)$ with the Hausdorff metric. Fix a compact subset $K\subset \in \mathcal{K}(X)$ and consider the map $$f_K: \mathcal{K}(X) \rightarrow \mathcal{K}(X), \qquad f_K(A) := A\cap K$$ We can turn $\mathcal{K}(X)$ into a measurable space by endowing it with the Borel $\sigma$-algebra $\mathcal{B}(\mathcal{K}(X))$.
Question. Is the map $$f_K: (\mathcal{K}(X), \mathcal{B}(\mathcal{K}(X))) \rightarrow (\mathcal{K}(X), \mathcal{B}(\mathcal{K}(X)))$$ measurable?