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  "body": "\n## Statement\n\nLet $(X, \\mathcal{A}, \\mu)$ be a measure space with $\\mu(X) \u003c \\infty$, that is, a [[finite measure|63Z1Q4FS23]] space.\nLet $T \\colon X \\to X$ be a [[measure-preserving transformation|556WBDAHNP]], and let $A \\in \\mathcal{A}$.\nThen for [[almost every|RARTMA0ZEC]] $x \\in A$, the point $T^k(x)$ lies in $A$ for infinitely many integers $k \\ge 1$.\n\n## Remarks\n\nLet $B$ be the set of points of $A$ that never return to $A$, so $B = A \\setminus \\bigcup_{k \\ge 1} T^{-k}(A)$.\nThe sets $B, T^{-1}(B), T^{-2}(B), \\dots$ are pairwise disjoint and all have measure $\\mu(B)$.\nSince their total measure is at most $\\mu(X) \u003c \\infty$, one gets $\\mu(B) = 0$.\nApplying this to $T^m$ for each $m \\ge 1$ shows that almost every point of $A$ returns infinitely often.\nThe hypothesis $\\mu(X) \u003c \\infty$ is needed: the translation $T(x) = x + 1$ on $\\R$ preserves Lebesgue measure, but no point of $A = [0,1)$ ever returns to $A$.\nExample: for an irrational rotation of the circle and an arc $A$ of positive length, every point of $A$ comes back to $A$ infinitely often.\n\n## Explanations\n\n### L1 human\n\nA system that keeps its total mass and lives in a bounded space must come back close to where it started.\nMore precisely, almost every point of a set of states returns to that set again and again.\nFor a gas in a closed box, this means the gas will, in principle, come back near any earlier state, though the waiting time is enormous.\n\n### L2 human\n\nThe theorem uses only finiteness of the measure and invariance, not any mixing property.\nIt is the starting point of recurrence in ergodic theory, which leads to multiple recurrence and Furstenberg's proof of Szemerédi's theorem.\nFor infinite measures, the maps for which the conclusion holds are called conservative, and the Mathlib form of the theorem is stated for conservative maps.\n",
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