Let $X$ be a compact Hausdorff space and fix a net of complex Borel measures $(\mu_n)$ such that $\mu_n \to \mu$ in the weak* (aka vague) topology on $C(X)^*$. That is, for every continuous function $f \colon X \to \mathbb{C}$ we have $\int_X f d\mu_n \to \int_X f d\mu$.
Is it true that $|\mu_n - \mu|(X) \to 0$, where $|\mu_n - \mu|$ is the total variation of $\mu_n - \mu$? If so, why?
*I'm taking the definition of total variation to be the one in Rudin - Real & Complex Analysis 3rd Ed, p116.
No, it is not true. For a counter example consider $X=[0,1]$ and let $\mu_n=\delta_{1/n}$, namely the Dirac measure giving full mass to the singleton $\{1/n\}$.
It is then easy to see that $\mu_n\to \delta_0$, and yet the total variation of $\mu_n-\delta_0$ is 2, for every $n$.