Let be a probability space and let $f\colon X\to X$ be a measure preserving transformation.

Theorem 1   For any , the set of those points $x$ of $E$ such that $f^n(x)\notin E$ for all $n>0$ has zero measure. That is, almost every point of $E$ returns to $E$ . In fact, almost every point returns infinitely often; i.e. $$\mu\left(\{x\in E:\textnormal{ there exists } N \textnormal{ such that }f^n(x)\notin E \textnormal{ for all } n>N\}\right)=0.$$

The following is a topological version of this theorem:

Theorem 2   If $X$ is a second countable Hausdorff space and contains the Borel sigma-algebra, then the set of recurrent points of $f$ has full measure. That is, almost every point is recurrent.