= Solution
For each $n\geq0$, choose an open arc $I_n$ about $e^{in}$ of arclength $\pi2^{-n-2}$, and put $G^1=\bigcup_{n\geq0}I_n$. It contains every point used by every empirical <measure>, whereas
$$
|G^1|\leq\sum_{n=0}^{\infty}\pi2^{-n-2}=\frac\pi2<\frac{|\mathbb T|}2.
$$
Thus \b[$\mu_N^1(G^1)=1$ for all $N$]. Let $K=\mathbb T\setminus G^1$. This is closed and $\mu_N^1(K)=0$ for all $N$. But $1/\pi$ is irrational, so the weak-star limit is normalized arclength and
$$
\boxed{\mu^1(K)=1-\frac{|G^1|}{2\pi}\geq\frac34>0.}
$$
Therefore the proposed implication is \b[false]. Weak-star convergence tests <continuous functions> and does not assert convergence of the <indicator function> of an arbitrary closed set.
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