Let $G$ be a group and let $F_i$ be a sequence of finite subsets of $G$. Suppose $G$ acts on a probability measure space $(X,\mu)$ in a measure preserving way, and suppose that this action is ergodic.
Let us say that $F_i$ satisfies pointwise ergodic theorem iff for almost all $x\in X$, and all $f\in L^1(X)$ we have that the limit of $$ \frac{1}{|F_i|} \sum_{g\in F_i} f(g.x) $$ is equal to $\int_X f\, d\mu$.
Let us say that $F_i$ satisfies mean sojourn time time iff for every measurable $U\subset X$ and almost every $x\in X$ we have that the limit of $$ \frac{1}{|F_i|} |\{g\in F_i\colon\, g.x \in U\}| $$ is equal to $\mu(U)$.
It is easy to see that if $F_i$ satisfies pointwise ergodic theorem then it also satisfies mean sojourn time theorem. Is it also the other way around? A reference would be most appreciated (I imagine that the answer in the general case is the same as in the case when $G$ is the infinite cyclic group, so a reference for the latter case would also be fine)
It is now answered at Math Overflow.