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

Given any amenable group $G$ (with a left Haar measure $|\cdot|$ or $dg$), we can select out a \textit{Følner subnet} $\{F_θ,θ\inΘ\}$ from any left Følner net in $G$, which is \textit{$L^\infty$-admissible}, namely, for any Borel $G$-space $(X,\mathscr{X})$ and any $φ\in L^\infty(X,\mathscr{X})$, \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 $σ$-compact such as a locally compact second countable Hausdorff amenable group, then $φ^*\in L^\infty(X,\mathscr{X})$, $φ^*(gx)=φ^*(x)$ \textit{a.e.}, and $φ^*$ is \textit{a.e.} independent of the choice of the admissible Følner net $\{F_θ,θ\inΘ\}$ in $G$. Consequently, we may easily obtain the ergodic disintegration of invariant probability measures for any $σ$-compact amenable group acting Borel on a compact metric space $X$ by continuous transformations of $X$, and the existence of $σ$-finite invariant Radon measures for any Borel action of an amenable group on a locally compact, $σ$-compact, metric space $X$ by continuous maps of $X$, and a $L^\infty$-pointwise multiple ergodic theorem.