= Wiener Fourier coefficient lemma
{c}
{title2=$\lim_{N\to\infty}N^{-1}\sum_{n<N}|\widehat\sigma(n)|^2=\sum_t\sigma(\{t\})^2$}
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>.
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