Let $X$ be an open bounded set in Euclidean space and $M(X) = C_c(X)^*$ be the space of signed Radon measures.
For $\mu_n$, $\mu \in M(X)$, by definition of vague convergence, $\mu_n \to \mu$ if and only if $\int_X f d\mu_n \to \int_X f d\mu$ for all compactly supported $f \in C_c(X)$.
Now, suppose that $\mu_n$ and $\mu$ are both positive and that $\mu_n \to \mu$ vaguely. I know that $\mu(X) < \infty$, but can I conclude that $\sup_n \mu_n(X) < \infty$?
The naive idea is to take $f(x) = 1$ everywhere but of course, $f$ does not have compact support. Then I was thinking taking a sequence of positive functions $f_m$ that approximates $f$ from below, but I couldn't justify how to exchange the limits $\lim_{n\to \infty}$ and $\lim_{m \to \infty}$.
$X=(0,1), \mu=n\delta_{1/n}, \mu=0$ is a counter-example.