# Is the average first return time of a partitioned ergodic transformation just the number of elements in the partition?

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.

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I found a 2003 paper of Choe containing your sanity check, called "A universal law of logarithm of the recurrence time". 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.