Inspired by an extension of Wiener's lemma on the relation of measures μ on the unit circle and their Fourier coefficients $\hat{\mu }(k_n)$ along subsequences $(k_n)$ of the natural numbers by Cuny, Eisner and Farkas [1], 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 [6] on the action of semigroups on Hilbert spaces.

受到维纳引理的扩展的启发，它关于单位圆上的度量μ及其傅里叶系数的关系$\hat{\mu }（{ķ}_{ñ}）$ 沿子序列 $（{ķ}_{ñ}）$根据Cuny，Eisner和Farkas [1]的自然数，我们研究了通过一系列概率测度对Fourier系数加权时引理的有效性。通过对从这些度量序列得出的滤波器使用收敛，我们获得了相似的结果，现在还可以考虑除$\mathbb{Ť}$ 和 $\mathbb{[R}$。作为应用，我们提出了Goldstein [6]结果对半群在希尔伯特空间上的作用的扩展。