= Solution
<Convolution> preserves <probability measures>: its defining pushforward is nonnegative and its total mass is the product of the two unit masses. For every fixed integer $n$ the previous coefficient identity gives
$$
\widehat{\mu_k*\nu_k}(n)=\widehat\mu_k(n)\widehat\nu_k(n)\longrightarrow\widehat\mu(n)\widehat\nu(n)=\widehat{\mu*\nu}(n).
$$
The <probability> version of the Fourier coefficient criterion therefore proves
$$
\boxed{\mu_k*\nu_k\overset{*}{\rightharpoonup}\mu*\nu.}
$$
The fixed compact underlying group and the uniform <probability> masses supply precisely the bounds needed in the dense-test argument.
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