= Solution
Let $m$ be normalized arclength on the <unit circle>, so in angular coordinates $dm=dt/(2\pi)$. It is a finite <Radon measure>. The given identity says that $\mu$ and $m$ represent the same <continuous linear functional> on $C(\mathbb T)$. Uniqueness in the <Riesz-Markov-Kakutani representation theorem> gives
$$
\boxed{\mu(E)=m(E)=\frac{|E|}{2\pi}\quad\text{for every Borel }E\subseteq\mathbb T.}
$$
The printed phrase “<Borel sets> in $E$” is interpreted as Borel subsets $E$ of the circle.
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