Let $X = [0,1]$. Let $\mu_n$ be a sequence of regular signed Borel measures on $X$, which converges to a measure $\mu$ on $X$ in weak-star, i.e. for any $f\in C_0(X)$, we have $\int_X f \mu_n(dx) \to \int_X f \mu(dx)$.
Does this imply that the total variation norm (=total mass) $|\mu|(X)$ converge?
That means: We know $\mu= \mu^+ - \mu^-$ (Jordan decomposition, $\mu^+$ and $\mu^-$ positive measures...). Then we defined $|\mu|(X)=\mu^+(X) - \mu^-(X)$.
Other formulation of question: Does then $|\mu_n|(X) \to |\mu|(X)$ ?