A-ergodicity of probability measures on locally compact groups

Abstract

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.