Uniform convergence of Lipschitz functions to characteristic function of a compact set

Question

Consider $(X,d)$ a metric space and $K \subseteq X$ a compact subset. I am trying to build a sequence of Lipschitz functions $f_n : X \to \mathbb R$ s.t. $f_n \to \chi_K$ uniformly.

If we try to build $$ f_n(x) := 1 - \max \{n d(x,K), 1 \} ,$$ we don't have uniform convergence because we have problems just outside the boundary of $K$ (more specifically, we can find $x \not \in K $ s.t. $d(x,K)$ is arbitrarily small, thus $ \sup _n \sup_{x \not \in K} |1 - \max \{ n d(x,K), 1 \} | = 1$).

Is there some sequence of Lipschitz functions which converges uniformly to $\chi_K$, or does the kind of behaviour outlined before always happen?

Answer 1

If $f_n$ converge uniformly, then the limit (in this case $\chi_K$) will be continuous. In most cases, this is false (for example for $K \neq \emptyset$ and $X = \Bbb{R}^n$).

To see this, note that (by the intermediate value theorem) $\chi_K$ would have to attain the value $1/2$ if it was continuous.

Answer 2

The uniform limit of a sequence of continuous functions is always continuous, so this will only work if the characteristic function of the compact set is continuous itself. However, this is in general not the case.

I assume you try to find such a sequence in order to prove something related to measure theory. It is probably not necessary to have uniform convergence, so it might be the best if you start a new question stating your original problem.