For some reason my thinking is *very* fuzzy today, so I apologize for the following rather silly question below...


Let $T$ be an ergodic transformation of $(X,\Omega, \mathbb{P})$ and let $X$ be partitioned into $n < \infty$ disjoint sets $R\_j$ of positive measure. For $x \in R\_k$ define $\tau(x) := \inf \{\ell>0:T^\ell x \in R\_k\}$. The Kac lemma (see, e.g. <http://arxiv.org/abs/math/0505625>) gives that $\int\_{R\_k} \tau(x) \ d\mathbb{P}(x) = 1$. 
Now $\int\_X \tau(x) \ d\mathbb{P}(x) = \sum\_k \int\_{R\_k} \tau(x) \ d\mathbb{P}(x) = n$, or equivalently $\mathbb{E}\tau = n$.


**Can anyone provide a sanity check on the above assertion that the expected return time is just the size of the partition?** I've never seen this explicitly stated as a corollary of the Kac lemma, which seems odd.



Is the average first return time of a partitioned ergodic transformation just the number of elements in the partition?
I found a 2003 paper of Choe containing your sanity check, called "[A universal law of logarithm of the recurrence time](http://iopscience.iop.org/0951-7715/16/3/306?ejredirect=migration)". See the first few lines of section 3 on page 888. The "$K\_n$" used there is essentially your $\tau$, but corresponding to a partition $P\_n$ in a sequence of partitions. See definition 1.7 on page 885 for what the notation means.


