Wiener Fourier coefficient lemma (${\lim }_{N\to \infty }{N}^{-1}{\sum }_{n<N}|\hat{\sigma }(n){|}^2={\sum }_t\sigma (\{t\}{)}^2$)

For a finite positive Borel measure on the circle, averaged squared Fourier coefficients tend to the sum of squares of its atomic masses. Express each squared coefficient as a double integral and average the circle character. The averaged character is bounded by one, tends to one on the diagonal and to zero elsewhere, so the dominated convergence theorem gives the diagonal mass. In particular an atomless measure has vanishing averaged absolute coefficients by the Cauchy-Schwarz inequality.