Archiv der Mathematik ( IF 0.6 ) Pub Date : 2023-11-29 , DOI: 10.1007/s00013-023-01938-y Heybetkulu Mustafayev
Let G be a locally compact group with the left Haar measure \(m_{G}\) and let \(A=\left[ a_{n,k}\right] _{n,k=0}^{\infty }\) be a strongly regular matrix. We show that if \(\mu \) is a power bounded measure on G, then there exists an idempotent measure \(\theta _{\mu }\) such that
$$\begin{aligned} \text {w*-}\lim _{n\rightarrow \infty }\sum _{k=0}^{\infty }a_{n,k}\mu ^{k}=\theta _{\mu }. \end{aligned}$$If \(\mu \) is a probability measure on a compact group G, then
$$\begin{aligned} \text {w*-}\lim _{n\rightarrow \infty }\sum _{k=0}^{\infty }a_{n,k}\mu ^{k}=\overline{m}_{H}, \end{aligned}$$where H is the closed subgroup of G generated by \(\text{ supp }\mu \) and \( \overline{m}_{H}\) is the measure on G defined by \(\overline{m}_{H}\left( E\right) :=m_{H}\left( E\cap H\right) \) for every Borel subset E of G.
中文翻译：
局部紧群概率测度的遍历性
设G为局部紧群，其左 Haar 测度\(m_{G}\)并设\(A=\left[ a_{n,k}\right] _{n,k=0}^{\infty }\)是一个强正则矩阵。我们证明，如果\(\mu \)是G上的幂有界测度，则存在幂等测度\(\theta _{\mu }\)使得
$$\begin{对齐} \text {w*-}\lim _{n\rightarrow \infty }\sum _{k=0}^{\infty }a_{n,k}\mu ^{k}= \theta_{\mu}。\end{对齐}$$如果\(\mu \)是紧群G上的概率测度，则
$$\begin{对齐} \text {w*-}\lim _{n\rightarrow \infty }\sum _{k=0}^{\infty }a_{n,k}\mu ^{k}= \overline{m}_{H}, \end{对齐}$$其中H是由\(\text{supp }\mu \)生成的G的闭子群，而\(\overline{m}_{H}\)是由\(\overline{m}_定义的G上的测度{H}\left( E\right) :=m_{H}\left( E\cap H\right) \)对于G的每个 Borel 子集E。
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