Radon measures which converge to zero vaguely Can we find (signed) Radon measures $\{\mu_n\}_{n \geq 1}$ such that for every $f \in C_0(\mathbb{R})$ we have $\int f \; d\mu_n \longrightarrow 0$ as $n \longrightarrow \infty$, but there exists a measurable, bounded (not continuous) function $g$ which is compactly supported, such that $\lim_{n \rightarrow \infty} \int g \; d\mu_n \neq 0$?
This is easily seen to be the case if $g$ is allowed to not be compactly supported (take $\mu_n$ as the uniform distribution on $[0, n]$, and $g=1$), but the compact support part is throwing me off. 

Define $\mu_n := \delta_0 - \delta_{1/n}$. Your function $g$ should now be easy to find.
