Kac's Lemma in ergodic theory tells us that given an ergodic measure preserving transformation $T$ of a probability space $(X,\mathcal{B},\mu)$, and a measurable subset $A\in\mathcal{B}$ with $\mu(A)>0$, we have $\int\limits_Ar_A\,d\mu=1$ where $r_A(x)=\inf\{n\geq 1 \mid T^nx\in A\}$ is the first return to $A$.
This incorporates the idea that if a set has large measure, the return time would be small, and if the set has small measure, the return time would be large.
Now, suppose we have a measurable partition of $A$, say $P$, where $\mu(p)>0$ for all $p\in P$. Suppose we have an integrable (not necessarily first) return time $t_A:A\to \mathbb{Z}^+$ which is constant on partition elements.
Is there an analogue of Kac's lemma for this return time? A guess would be $\frac{1}{\mu(p)}\leq t_A(p)$ for all $p\in P$, but I am not sure if this is true. This does seem to capture the intuition given in the second paragraph.
Edit: It seems my guess can't be true, due to integrability of $t_A$. I can however show that $\mu(p)t_A(p)\leq \left\|t_A\right\|_1$. Is it possible to get anything better?
Thanks!