Let $X$ be a compact Hausdorff space and let $M(X)$ denote the space of signed measures that is naturally dual to $C(X)$, the space of continuous functions on $X$. I am interested whether the following condition:


$$\nu\_n(O) \to \nu(O)$$


for every open set $O\subset K$ is sufficient for weak\* convergence of $\nu\_n$ to $\nu$, at least when $X$ is zero-dimensional but not necessarily metrizable. Thank you.



Weak*-convergence of signed measures
The answer is positive if the sequence $(\nu\_n)$ is bounded; please see Theorem IV.9.15 in Dunford/Schwartz, vol. 1. PS: I just notice that this was already observed by Christian a couple of minutes ago.


