= Fourier distance of order two
{c}
{title2=$d_2(f,g)=\sup_{\xi\ne0}|\widehat f(\xi)-\widehat g(\xi)|/|\xi|^2$}
For finite nonnegative measures with equal mass and first moment and finite second moments, the constant and linear terms of their <Fourier transforms> cancel. Taylor's remainder bounds their difference by a constant times $|\xi|^2$, making this distance finite. Fourier uniqueness gives definiteness. The origin is excluded from the supremum; the quotient need not have a direction-independent limit there.
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