Pointwise ergodic theorem for amenable semigroups

Using tempered Følner sequences one may show a pointwise ergodic theorem for amenable groups.

Is there a similar result for semigroups? Where by taking a special kind of Følner sequence you can achieve this. (Either for $L^{1}$ or $L^{2}$)

• I think something like this appears in Krengel's book. – Anthony Quas Dec 20 '13 at 18:57

Let $\Gamma$ be a discrete, countable, left-cancellative, left-amenable semigroup acting on $(X,\mathcal{B},\mu)$ via measure-preserving maps. Let $(F_N)_N$ be a tempered Følner sequence in $\Gamma$. If the action of $\Gamma$ is ergodic, then for all $f \in L^1(X,\mathcal{B},\mu)$, $$\lim_{N \to \infty} \frac 1{|F_N|} \sum_{\gamma \in F_N} f(\gamma x) \longrightarrow \int f \ d\mu \qquad \mu\text{-a.e.}$$