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A-ergodicity of probability measures on locally compact groups

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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.

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The author is grateful to the referee for his helpful remarks and suggestions.

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Correspondence to Heybetkulu Mustafayev.

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Mustafayev, H. A-ergodicity of probability measures on locally compact groups. Arch. Math. 122, 47–57 (2024).

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