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{F{\o}lner subnet} $\{F_\theta,\theta\in\Theta\}$ from any left F{\o}lner net in $G$, which is \textit{$L^\infty$-admissible}, namely, for any Borel $G$-space $(X,\mathscr{X})$ and any $\varphi\in L^\infty(X,\mathscr{X})$, \begin{gather*} \lim_{\theta\in\Theta}\frac{1}{|F_\theta|}\int_{F_\theta}\varphi(gx)dg=\varphi^*(x)\ \forall x\in X\quad {\textrm{and}}\quad \varphi^*=(g\varphi)^*\ \forall g\in G. \end{gather*} Moreover, if $G$ is $\sigma$-compact such as a locally compact second countable Hausdorff amenable group, then $\varphi^*\in L^\infty(X,\mathscr{X})$, $\varphi^*(gx)=\varphi^*(x)$ \textit{a.e.}, and $\varphi^*$ is \textit{a.e.} independent of the choice of the admissible F{\o}lner 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.