  Inspired by an extension of Wiener's lemma on the relation of measures $\mu$
on the unit circle and their Fourier coefficients $\widehat{\mu}(k_n)$ along
subsequences $(k_n)$ of the natural numbers by Cuny, Eisner and Farkas [CEF19,
arXiv:1701.00101], we study the validity of the lemma when the Fourier
coefficients are weighted by a sequence of probability measures. By using
convergence with respect to a filter derived from these measure sequences, we
obtain similar results, now also allowing the consideration of locally compact
abelian groups other than $\mathbb{T}$ and $\mathbb{R}$. As an application, we
present an extension of a result of Goldstein [Gol96] on the action of
semigroups on Hilbert spaces.
