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  "body": "\n## Definition\n\nLet $(X, \\mathcal{A}, \\mu)$ be a measure space with $\\mu$ [[σ-finite|WEBBT18D4X]], and let $T \\colon X \\to X$ be a [[quasi-measure-preserving map|F608XX51KA]], with [[iterates|AYVTR2ES5B]] $T^n$.\nThe map $T$ is *conservative* if for every $A \\in \\mathcal{A}$ with $\\mu(A) \u003e 0$ there is an integer $n \\ge 1$ with\n$$\\mu(A \\cap T^{-n}(A)) \u003e 0.$$\n\n## Remarks\n\nEquivalently, $T$ has no wandering set of positive measure, where $W \\in \\mathcal{A}$ is wandering if the sets $T^{-n}(W)$, $n \\ge 0$, are pairwise disjoint.\nFor a conservative $T$, almost every point of each $A \\in \\mathcal{A}$ returns to $A$ infinitely often.\nThe Hopf decomposition splits $X$, up to null sets, into an invariant part on which $T$ is conservative and a part that is a union of wandering sets.\n\nExample: every measure-preserving map of a finite measure space is conservative, by the [[Poincaré recurrence theorem|TVYJPBHD0S]].\nFor instance, an irrational rotation of the circle with Lebesgue measure is conservative.\n\nNon-example: the translation $T(x) = x + 1$ on $\\R$ with Lebesgue measure preserves the measure, but for $A = [0, 1)$ one has $T^{-n}(A) = [-n, 1 - n)$, which is disjoint from $A$ for every $n \\ge 1$.\nSo $\\mu(A \\cap T^{-n}(A)) = 0$ for all $n \\ge 1$, and $T$ is not conservative.\n",
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