I'm stuck on exercise (1.2.2) of Foundations of Ergodic Theory by Viana and Oliveira. The claim is this:
Let f : M → M be an invertible transformation and suppose that µ is an invariant measure, not necessarily finite. Let B ⊂ M be a set with finite measure. Prove that, given any measurable set E ⊂ M with positive measure, µ-almost every point x ∈ E either returns to E an infinite number of times or has only a finite number of iterates in B.
Here's what I have so far. Wlog, we can assume that every element of B recurs to B infinitely often, and $\mu(B) > 0$. Define the set $$E_0 = \{x\in E \, | \, f^n x \notin E \text{ for all } n\geq 1 \text{ and } \exists n f^n x \in B\}.$$ It suffices to show that $\mu(E_0) = 0.$ It is clear that the sets $$E_0, f^{-1}E_0, \ldots$$ are disjoint. On the other hand, for infinitely many $n$, $f^{-n}B \cap E_0 \neq \emptyset.$ Therefore, there is an $n$ such that $$\mu(f^{-n} B \cap E_0) > 0.$$ Let $n$ be such an integer, and define $F = f^{-n}B \cap E_0$. Then $F, f^{-1}F,f^{-2} F, \ldots$ are disjoint sets of measure $\mu(F) > 0$. Therefore,$$\mu(\bigcup f^{-n} F) = \sum \mu(f^{-n} F) = \infty.$$
At this point I'm stuck. Does anyone have a hint on how to proceed? I am also not sure how to use the fact that $f$ is invertible. Is the invertible assumption necessary?
Update I believe I have a solution, but I'm still wondering if the invertible assumption is necessary. Here's how we can finish the proof. Since f is invertible, $f(A_1 \cap A_2) = f(A_1) \cap f(A_2)$ for all sets $A_1,A_2$. Therefore, since they are disjoint, for every $k$, $$f^{n+k+1}(f^{-k}F \cap f^{-k-1}F) = \emptyset.$$ By definition, this means that $$f^k(B \cap f^nE_0) \cap (B \cap f^n E_0) = \emptyset,$$ for every $k \geq 1$. Since every element of $B$ recurs infinitely often, this means that the disjoint sets $f^k(B \cap f^nE_0)$ cover $B$. However, these sets have the same measure, and we have a contradiction. Note that this assumes that $f^k(B \cap f^nE_0)$ is measurable...