Gaussian positivity for even extension moments

Let $\mu$ be a finite positive surface measure and $|f|\le 1$. Expand $|\widehat{f\mu }(x){|}^{2k}$ and integrate against $e^{-\pi |x{|}^2/R^2}$. The Fourier transform of a Gaussian turns the oscillatory kernel into $R^d{e}^{-\pi R^2|{\sum }_{j\le k}{\omega }_j-{\sum }_{j>k}{\omega }_j{|}^2}$, which is nonnegative. Taking absolute values of the bounded amplitudes gives an upper bound by the same moment with $f=1$. This weighted positivity is valid even when $f$ is merely measurable; it transfers a scalar surface-decay estimate to a local bounded-density moment estimate.