Pointwise convergence of ergodic averages of bounded measurable functions for amenable groups

Abstract

Given any amenable group $G$ (with a left Haar measure $|\cdot |$ or $dg$), we can select out a \textit{Følner subnet} $\{F_{\theta },\theta \in \Theta \}$ from any left Følner net in $G$, which is \textit{$L^{\infty }$-admissible}, namely, for any Borel $G$-space $(X,\mathcal{X})$ and any $\phi \in L^{\infty }(X,\mathcal{X})$, \begin{gather*} \lim_{θ\inΘ}\frac{1}{|F_θ|}\int_{F_θ}φ(gx)dg=φ^*(x)\ \forall x\in X\quad {\textrm{and}}\quad φ^*=(gφ)^*\ \forall g\in G. \end{gather*} Moreover, if $G$ is $\sigma$-compact such as a locally compact second countable Hausdorff amenable group, then ${\phi }^{\ast }\in L^{\infty }(X,\mathcal{X})$, ${\phi }^{\ast }(gx)={\phi }^{\ast }(x)$ \textit{a.e.}, and ${\phi }^{\ast }$ is \textit{a.e.} independent of the choice of the admissible Følner net $\{F_{\theta },\theta \in \Theta \}$ in $G$. Consequently, we may easily obtain the ergodic disintegration of invariant probability measures for any $\sigma$-compact amenable group acting Borel on a compact metric space $X$ by continuous transformations of $X$, and the existence of $\sigma$-finite invariant Radon measures for any Borel action of an amenable group on a locally compact, $\sigma$-compact, metric space $X$ by continuous maps of $X$, and a $L^{\infty }$-pointwise multiple ergodic theorem.

33 pages

References in corpus (1)

• Pointwise convergence of multiple ergodic averages and strictly ergodic models