= Spherical derivative of a singular measure vanishes
{title2=$\mu\perp\lambda_d\Longrightarrow D_s\mu=0\quad\lambda_d\text{-a.e.}$}
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.
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