Let $K$ be a compact Hausdorff space (not necessarily metric or even separable). Let $M(K)$ be the space of all Radon measures on $K$ (that is, finite signed regular Borel measures) endowed with the variation norm. By the Riesz representation theorem $M(K)$ is isometrically isomorphic to the dual space $C(K)^*$ of the space $C(K)$ of all continuous real-valued functions on $K$ endowed with the supremum norm. By the weak topology on $M(K)$ I mean the topology given by the subbase consisting of the sets $V(\mu;x^*;\varepsilon)=\{\nu\in M(K)\colon\ |x^*(\mu)-x^*(\nu)|<\varepsilon\}$, where $\mu\in M(K)$, $x^*\in M(K)^*$ ($\cong C(K)^{**}$), and $\varepsilon>0$.
Let $(\mu_n)$ be a sequence in $M(K)$. It is well-known that $(\mu_n)$ converges to some $\mu\in M(K)$ with respect to the weak topology if and only if $\lim_{n\to\infty}\mu_n(B)=\mu(B)$ for every Borel set $B\subseteq K$. My question is thus the following:
QUESTION: Can we define the weak topology on $M(K)$ by means of the subbase given by the sets of the form $W(\mu;B;\varepsilon)=\{\nu\in M(K)\colon\ |\mu(B)-\nu(B)|<\varepsilon\}$, where $\mu\in M(K)$, $B\subseteq K$ Borel, and $\varepsilon>0$?
(If it helps, one may assume that $K$ is totally-disconnected.)
