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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.

The Fourier coefficient of a measure on the circle and convolution of measures on the circle use the angular group law: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,Thus $\widehat{\mu \ast \nu }(n)=\hat{\mu }(n)\hat{\nu }(n)$. These measure coefficients use no additional Haar normalization factor: a probability has zeroth coefficient one.