On composition convergent sequences of measures on compact groups

A seguence of measures $\{\mu_n\}$ on a compact group $G$ is called composition convergent if, for any $i$, the seguence of measures $\mu_i\mu_{i+1}\dots\mu_{i+n}$ converges as $n\to\infty$ in the ordinary sense to a measure $\nu_i$. The main property of composition convergent sequences is that a limit of $\nu_i$ as $i\to\infty$ exists and is equal to the uniform measure on a subgroup $g$, $g\subseteq G$ which we call the base of $\{\mu_n\}$.
We show that for any sequence $\{\mu_n\}$ there corresponds a composition convergent sequence $\{\mu'_n\}$ obtained from $\{\mu_n\}$ by shifts: $\mu'_n=\alpha^{-1}_{n-1}\mu_n\alpha_n$, $\alpha_n\in G$. This fact enables to make use of the above property of composition convergent sequences of measures when dealing with convergence conditions for compositions of different measures $\{\mu_n\}$ on $G$.