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Given any amenable group $G$ (with a left Haar measure $|\cdot|$ or $dg$), wecan select out a \textit{F{\o}lner subnet} $\{F_\theta,\theta\in\Theta\}$ fromany left F{\o}lner net in $G$, which is \textit{$L^\infty$-admissible}, namely,for any Borel $G$-space $(X,\mathscr{X})$ and any $\varphi\inL^\infty(X,\mathscr{X})$, \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 compactsecond countable Hausdorff amenable group, then $\varphi^*\inL^\infty(X,\mathscr{X})$, $\varphi^*(gx)=\varphi^*(x)$ \textit{a.e.}, and$\varphi^*$ is \textit{a.e.} independent of the choice of the admissibleF{\o}lner net $\{F_\theta,\theta\in\Theta\}$ in $G$.  Consequently, we may easily obtain the ergodic disintegration of invariantprobability measures for any $\sigma$-compact amenable group acting Borel on acompact metric space $X$ by continuous transformations of $X$, and theexistence of $\sigma$-finite invariant Radon measures for any Borel action ofan amenable group on a locally compact, $\sigma$-compact, metric space $X$ bycontinuous maps of $X$, and a $L^\infty$-pointwise multiple ergodic theorem.

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