= Solution
The <Fourier coefficient of a measure on the circle> and <convolution of measures on the circle> use the angular group law:
$$
\widehat\mu(n)=\int_{\mathbb T}e^{-inx}\,d\mu(x),\qquad
(\mu*\nu)(E)=\int_{\mathbb T}\int_{\mathbb T}\mathbf1_E(x+y)\,d\mu(x)d\nu(y).
$$
Addition is modulo $2\pi$. This is the pushforward of the product <complex measure> under addition; equivalently its integral against a continuous $f$ is $\iint f(x+y)d\mu(x)d\nu(y)$. The integrands are bounded and the variation <measures> finite, so Fubini applies. In particular,
$$
\widehat{\mu*\nu}(n)=\iint e^{-in(x+y)}d\mu(x)d\nu(y)=\left(\int e^{-inx}d\mu(x)\right)\left(\int e^{-iny}d\nu(y)\right).
$$
Thus \b[$\widehat{\mu*\nu}(n)=\widehat\mu(n)\widehat\nu(n)$]. These <measure> coefficients use no additional Haar normalization factor: a <probability> has zeroth coefficient one.
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