Let $(X,\parallel \cdot \parallel)$ be Banach and $$\mathcal{BC}(X)=\{A\subset X\colon A \text{ is closed, bounded and non-empty}\}.$$ The natural metric on this space is the Hausdorff distance $d_H$ (see http://en.wikipedia.org/wiki/Hausdorff_distance)
Let $C_x(r)$ be the closed ball around $x\in X$ with radius $r$. How can I show that the map $f\colon X\to\mathcal{BC}(X)$ where $f(x)=C_x(r)$ is continuous w.r.t. $d_H$?
And is the map $g\colon X\to\mathbf{R}$ with $g(x)=\mu(C_x(R))$ Borel measurable for a Borel measure $\mu$?