Want to find $f\_n$ a sequence of continuous functions, so that for all Borel regular measure $\mu$, we have 


$\int f\_n d\mu\rightarrow\int \chi\_\Delta d\mu$, as $n$ goes to infinity, where $\Delta$ is a Borel set.



Can the characteristic function of a Borel set be approached by a sequence of continuous function through a certain convergence in $L^\infty$?
(EDITED)
Let $\Delta$ be the rationals in $[0,1]$. If your condition is satisfied, in particular $f\_n$ converges pointwise to $\chi\_\Delta$. Let $C\_n = \{x \in [0,1]: \forall m > n, \;|f\_n(x) - f\_m(x)|\le 1/3\}$. Then $C\_n$ are closed and their union is $[0,1]$. By the Baire category theorem some $C\_n$ has nonempty interior. But this is impossible since $f\_n$ is continuous and both $\Delta$ and its complement are dense.


