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.
