Kakutani skyscraper is infinite

Question

Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56

Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, \mathcal B, \mu)$ there exists a set $A$ of positive measure for which the return time $n_A$ is unbounded.

Notation: Here $n_A$ is given by $n_A (x):= \min\{n: T^n(x)\in A \} $, so $n_A: A \rightarrow \mathbb N$ is an a.e. defined measurable map (by Poincare Recurrence) and it is the smallest integer such that the point $x\in A$ returns to $A$ under the action of $T$. By unbounded I suppose he means that the essential supremum of $n_A$ is not finite...

I could really use some help here! Thanks so much

Answer 1

Note: Petersen assumes that $T$ is bijective a.e. (p.2)

This one is a bit tricky, but I think the following intuition can be made to work: We construct a gasket of positive measure by deleting countably many sets from $X$, leaving a set of positive measure with unbounded return time.

With the Kakutani-Rokhlin Lemma (Lemma 4.7, p. 48) we can find some set $A$ with measure less than a half, such that $TA$ and $A$ are disjoint. Delete $TA$ from $X$, then the essential supremum of $n_{X/TA}$ is at least $2$ (on $A$) and $\mu(X/TA)>1/2$.

Looking at the skyscraper of how $A$ was constructed in the Kakutani-Rokhlin Lemma, it should be possible to construct a sufficiently small set of positive measure $B\subset A$ with $TB,T^3B\subset TA$ and $T^2B\subset A$ and $B,TB,T^2B,T^3B$ disjoint. Delete $T^2B$ from $X/TA$, so that the resulting set has return time at least $4$ on B.

Continuing in this way, with successive (not too large) deletions from $X$, should yield the set we're looking for. I'll leave the exact details.