pr.probability - 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 $int_{R_k} tau(x) dmathbb{P}(x) = 1$.
Now $int_X tau(x) dmathbb{P}(x) = sum_k int_{R_k} tau(x) dmathbb{P}(x) = n$, 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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