Originally posted on Maths StackExchange, but repositing here because of getting no answer there. Not a research question really - I'm just confused by implications between various ergodic theorems. So I'll happily close the question if deemed inappropriate.

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
$$
\frac{1}{|F_i|} \sum_{g\in F_i} f(g.x)
$$
exists and is equal to $\int_X f\, d\mu$.

Let us say that $F_i$ *satisfies mean sojourn time theorem* iff for every measurable $U\subset X$ and almost every $x\in X$ we have that the limit of
$$
\frac{1}{|F_i|} |\lbrace g\in F_i\colon g.x \in U\rbrace |
$$
exists and is equal to $\mu(U)$.

Question 1: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.)

A related question:

Question 2:Is there a proof of the mean sojourn theorem for say $\mathbb Z$ which doesn't use the pointwise ergodic theorem?