Let $f:M\to M$ be a transformation and $\mu $ a finite invariant measure for $f$. Let $E$ be a subset of $M$ with positive measure^{}. We define the first return map for $E$:

$${\rho}_{E}(x)=\mathrm{min}\{n\ge 1:{f}^{n}(x)\in E\}$$ 

If the set on the right is empty, then we define ${\rho}_{E}(x)=\mathrm{\infty}$. The PoincarÃ© recurrence theorem asserts that ${\rho}_{E}$ is finite for almost every $x\in R$.
We define the following sets:

$${E}_{0}=\{x\in E:{f}^{n}(x)\notin E,n\ge 1\}$$ 


$${E}_{0}^{*}=\{x\in M:{f}^{n}(x)\notin E,n\ge 0\}$$ 

By PoincarÃ© recurrence theorem, $\mu ({E}_{0})=0$.
Kac’s theorem asserts that the function ${\rho}_{E}$ is integrable and

$${\int}_{E}{\rho}_{E}\mathit{d}\mu =\mu (M)\mu ({E}_{0}^{*})$$ 

When the system is ergodic, then $\mu ({E}_{0}^{*})=0$, and Kac’s theorem implies:

$$\frac{1}{\mu (E)}{\int}_{E}{\rho}_{E}\mathit{d}\mu =\frac{\mu (M)}{\mu (E)}$$ 

This equality can be interpreted as: the mean return time to $E$ s inversely proportional to the measure of $E$.