Let $(X,d)$ be a compact separable metric space. Let $\mu$ be a Borel, regular, finite, **signed** measure on $X$ such that for all $x\in X$, for all $r>0$, $\mu(B(x,r))=0$, where $B$ denotes the (either open or closed) ball w.r.t $d$.

Is $\mu$ zero?

If $\mu$ is positive one can show that $\mu=0$ using the Borel-Lebesgue theorem, but what if $\mu$ is signed?