Let $X$ be Banach and let $B(x,\varepsilon)$ be the closed ball of radius $\varepsilon>0$ around $x\in X$ and consider the sequence $f_{n;x}(y)= \begin{cases} 1-n\cdot d(yB(x,\varepsilon)), & (1-n\cdot d(y,B(x,\varepsilon)))\ge0\\ 0, &\text{elsewhere}. \end{cases} $ for $y\in X$ and $n=1,2,\ldots$ and where $d(y,B(x,\varepsilon))=\inf\{\parallel y-z\parallel\colon z\in B(x,\varepsilon)\}$. The pointwise limit of this sequence is the characteristic function on $B(x,\varepsilon)$.
Let $BC(X)$ denote the space of closed and bounded subset of $X$, then the map $f\colon X\to BC(X): x\mapsto B(x,\varepsilon)$ is continuous for the Hausdorff metric $d_H$ . And for $A\in BC(X)$ fixed, the map $g_A\colon BC(X)\to\mathbb{R}: B\mapsto\delta(A,B),$ is 1-Lipschitz, where $\delta(A,B)=\sup_{a\in A}\inf_{b\in B}\parallel a-b \parallel$.
Is there somebody who knows how to prove the following:
Let $\{x_n: n=1,2,\ldots\}$ be a convergent sequence in $X$ with limit $x_0$, then $f_{n,x_j}\to f_{n,x_0}$ pointwise as $j\to\infty$.
My main goal is to show that the map $p\colon X\to\mathbb{R}:x\mapsto\mu(B(x,\varepsilon))$ is Borel measurable for any (not necessarily finite or positive) Borel measure $\mu$ on $X$. If I would be able to prove $f_{n,x_j}\to f_{n,x_0}$, then I'm done.