The classical Kac's lemma says the following.

Let $(X,\mu)$ be a probability space and $T$ a measure preserving transformation. Assume $A\subset X$ has positive measure. Then $$\sum_{k\ge 1} k\mu(A_k)=1,$$ where $A_k$ denotes the set of points in $A$ that return to $A$ by first time after exactly $k$ iterates of $T$.

**Question:** Is there an 'analog' of Kac's lemma for amenable group actions? with suitably choice of folner sequence and dealing with the (a priori) lack of disjointness that the sets $A_k$'s propagate. Of course I am not expecting the (suitable) weigthed sum to have value 1, but something finite.

It sounds like a natural question and I am not sure if it's a known fact.