Weak convergence of measures on dense sets

Question

We are given a complete (separable) metric space $X$ and a dense subset $D\subset X$. Consider a sequence of continuous functions $f_n\colon X\to \mathbb R$ such that $$\int\limits_D f_n \, {\rm d}\mu\to 0$$ for every compactly supported finite signed Borel measure $\mu$ on $D$. It seems to me that it is asking for too much to have

$$\int\limits_X f_n \, {\rm d}\mu\to 0$$ for every compactly supported finite signed Borel measure $\mu$ on $X$ as $D$ may have much fewer compact sets than $X$, however I cannot find a counterexample.

Have such phenomena been investigated somewhere in the literature?

Answer 1

Take $X:=\mathbb{R}$ and $D:=\mathbb{R}\setminus\{0\}$. Consider any sequence of continuous functions $(f_n)_n$ that converges uniformly to $0$ on compact sets of $D$, but with $\langle \delta_0,f_n\rangle:=f_n(0)=1$, like e.g. $f_n(x):=(1-n|x|)_+$ .

rmk. Of course the same example works for any $D\subset\mathbb{R}$ such that $0\notin D$, thus also, up to a translation, for any proper subset $D$ of $\mathbb{R}$.