Moscow Mathematical Journal
Volume 26, Issue 1, January–March 2026 pp. 1–17.
Mean Ergodic Theorem for Two Probability Measures on Compact Groups
Let $G$ be a locally compact group with the left Haar measure $m_{G}$. A
probability measure $\mu $ on $G$ is said to be strictly aperiodic
if the support of $\mu $ is not contained in a proper closed left cosets of $G$. Let $\mu $ and $\nu $ be two commuting (with respect to convolution)
probability measures on a compact group $G$ and let
\begin{equation*}
M_{n}( \mu ,\nu ) :=\frac{1}{n+1}\sum_{i=0}^{n}\mu ^{i}\ast \nu ^{n-i}
\end{equation*}
be the Birkhoff average of the pair $( \mu ,\nu ) $. Among other
results, we show that if one of these measures is strictly aperiodic, then
\begin{equation*}
\mathop{\text{w\(^{\ast }\)-\(\lim\)}}_{n\rightarrow \infty }M_{n}( \mu ,\nu ) =
\overline{m}_{H},
\end{equation*}
where $H$ is the closed subgroup of $G$ generated by $\operatorname{supp}\mu \cup \operatorname{supp}\nu
$ and $\overline{m}_{H}$ is the measure on $G$ defined by $\overline{m}_{H}( B) =m_{H}( B\cap H) $ for every Borel subset $B$ of $G$.
2020 Math. Subj. Class. 28A33, 43A10, 43A77, 47A35.
Authors:
Araz R. Aliev (1) and Heybetkulu S. Mustafayev (1)
Author institution:(1) Azerbaijan State Oil and Industry University, Baku-Azerbaijan;
Institute of Mathematics, Baku-Azerbaijan;
Center for Mathematics and its Applications, Khazar University,
Baku-Azerbaijan
Summary:
Keywords: Mean ergodic theorem, compact group, probability measure, convergence.
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