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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 $\mu$. Is it obvious that $B$ is a Borel set in $X$? If it is Borel, what is its Borel class? 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 $X$.

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  • $\begingroup$ I guess you can show that total variation is a measurable function, but not sure whether it will help to answer which Borel class does $B$ belong to. $\endgroup$
    – SBF
    Dec 5, 2014 at 16:28

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