On the spatial mean of the Poincare cycle

Abstract

Let $X$ be a measure space and $T:X\to X$ a measurable transformation. For any measurable $E\subseteq X$ and $x\in E$, the possibly infinite return time is $n_E(x):=\inf \{n>0:T^{n}x\in E\}$. If $T$ is an ergodic tranformation of the probability space $X$, and $\mu (E)>0$, then a theorem of M. Kac states that ${\int }_E{n}_{E}d\mu =1$. We generalize this to any invertible measure preserving transformation $T$ on a finite measure space $X$, by proving independently, and nearly trivially that for any measurable $E\subseteq X$ one has ${\int }_E{n}_{E}d\mu =\mu (I_E)$, where $I_E$ is the smallest invariant set containing $E$. In particular this also provides a simpler proof of Poincaré's recurrence theorem.

2 pages, Translation into English of a paper by the author generalizing Kac's theorem on the spatial mean of the Poincare cycle. Of possible pedagogical value