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.