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The spherical derivative compares the mass of shrinking centered open balls with their Lebesgue measure, when the limit exists. This is a density of one measure relative to Euclidean volume, rather than the relative density of a set within one fixed measure. A singular finite Borel measure has spherical derivative zero Lebesgue almost everywhere.

For a finite singular Borel measure, choose compact subsets of a Lebesgue-null carrier whose omitted mass is arbitrarily small. Outside any such compact subset, sufficiently small balls see only the omitted measure. The uncentered maximal weak-type inequality bounds the volume of points where the omitted measure has large density. Letting its total mass tend to zero shows that the upper spherical density is zero almost everywhere. This argument does not require the null carrier itself to be closed or separated from every point outside it.

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.