= Convolution of measures on the circle
{title2=$\widehat{\mu*\nu}(n)=\widehat\mu(n)\widehat\nu(n)$}
For finite <complex measures> on the additive <circle group>, <convolution> is the <pushforward measure> of their product under addition modulo $2\pi$. Equivalently
$$
\int f\,d(\mu*\nu)=\iint f(x+y)d\mu(x)d\nu(y).
$$
Bounded tests are integrable against the product variation <measure>, so <Fubini's theorem> applies. Testing the factorization $e^{-in(x+y)}=e^{-inx}e^{-iny}$ proves the displayed <Fourier coefficient of a measure on the circle> identity. The <convolution> of two <probability measures> is again a <probability measure>. On these <measures> <convolution> preserves simultaneous convergence in the <weak-star topology>: multiply the converging coefficients and use their common unit variation bound and dense trigonometric tests.
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