Definition of integrable return time

Question

Let $(X,\mathcal{B},\mu,T)$ be an ergodic measure preserving dynamical system and $Y\subset X$ with $\mu(Y)>0$.

The first return to $Y$ is the function $R:Y\to \mathbb{N}$ defined by $R(y)=\inf\{n\in \mathbb{N} \mid T^n(y)\in Y\}$. This can easily be extended to a $kth$ return time, where $k$ is any natural number.

In reading papers, I keep coming across the notion of "integrable return times". I can not seem to find any definition online (all ergodic theory notes refer to first returns), but if I was to hazard a guess, the following definition seems natural:

"An integrable return time is an integrable function $S:Y\to \mathbb{N}$ such that $T^{S(y)}\in Y$".

My question is: Is the above definition correct? If not, is anyone aware of the correct definition?

Answer 1

It means that $R$ is integrable on $Y$.

Then all $k$th return times are integrable. Actually, you can forget about "ergodic" in your first sentence. Nothing changes.

But you should not use your definition, not all such functions are integrable if $R$ is integrable (notice the $\inf$ in the definition) even if they are measurable.