= Solution
For a <complex measure>, its <total variation measure> is
$$
|\mu|(E)=\sup\left\{\sum_j|\mu(E_j)|:E=\bigsqcup_{j=1}^mE_j,\ E_j\text{ Borel}\right\}.
$$
Its norm is \b[$\|\mu\|=|\mu|(\mathbb T)$]. For <Radon measures> on the compact circle this is also $\sup_{\|f\|_\infty\leq1,f\in C(\mathbb T)}|\int f\,d\mu|$. The stated convergence is weak-star convergence against continuous tests:
$$
\boxed{\mu_k\overset{*}{\rightharpoonup}\mu\iff\int f\,d\mu_k\to\int f\,d\mu\quad\text{for every }f\in C(\mathbb T).}
$$
It is the <weak-star topology> from the identification of these <measures> with the dual of $C(\mathbb T)$, rather than convergence of their values on every <Borel set>.
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