Convolution of measures on the circle ($\widehat{\mu \ast \nu }(n)=\hat{\mu }(n)\hat{\nu }(n)$)

For finite complex measures on the additive circle group, convolution is the pushforward measure of their product under addition modulo $2\pi$. EquivalentlyBounded 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.