Let $K$ be a compact Hausdorff space and consider $X=Ball(M(K))$, the unit ball of the space of regular Borel measures on $K$. Endow $X$ with the weak-$*$ topology $\sigma(M(K),C(K))$, regarding $M(K)$ as the dual space of the space $C(K)$ of continuous scalar valued functions. The scalars can be either real or complex. Then $X$ is also a compact Hausdorff space. For a compact subset $F$ of $K$ and $\varepsilon >0$ consider the set $B=\{\mu\in X: |\mu|(F)<\varepsilon\}$, where $|\mu|$ is the total variation of $\mu$. What is the Borel class of $B$ as a subset of $X$? Is there a text reference for this?

The best I can do is to express $B$ as a countable union of sets of the form $F_i\cap G_i$ where the $F_i$ are closed sets in $X$ and the $G_i$ are open in $X$.