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There is a norm for signed measures given by $$||\mu||_{FM}=\sup_{f\in \mathrm{Lip}_1(X),|f(x)|\leq 1}\langle f,\mu\rangle.$$ This is usually called Fortet-Mourier norm (or more often metric, but it is a norm).

For probability measures weak (or weak-star) convergence of measures is equivalent to convergence in $||\cdot||_{FM}$. This equivalence can't happen however for the whole space of signed measures, cause weak-star topology in space $X^*$ is never metrizable if $X$ is infinitely dimensional. (And by Riesz's representation theorem $X=C_c(X)$ if $X^*$ are signed measures)

But it is pretty hard to find an example of sequence of signed measures that convergence in one of this topologies and doesn't in the second one. Can anybody show such a counterexample?

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Let's take $X=[0,1]$, closed unit interval. Let $\mu_n = \sqrt{n} (\delta_0 - \delta_{1/n})$, where the deltas are unit point masses. Then $\|\mu_n\|_{FM} = \sqrt{n} \frac{1}{n} \to 0$ as $n\to\infty$. However, $\mu_n$ does not converge weakly to $0$. For example, $f(x)=\sqrt{x}$ is in $C(X)$, and $\langle f,\mu_n\rangle = -1$ for all $n$.

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