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  "body": "\n## Statement\n\nLet $(X, \\mathcal{B}, \\mu)$ be a [[standard probability space|QDDRDW9PXS]] and let $T \\colon X \\to X$ be an invertible [[measure-preserving transformation|556WBDAHNP]] with measurable inverse.\nAssume that $T$ is aperiodic: for every $n \\ge 1$, the set $\\set{x \\in X : T^n x = x}$ has measure $0$.\nThen for every integer $n \\ge 1$ and every $\\varepsilon \u003e 0$ there is a set $B \\in \\mathcal{B}$ such that:\n\n1. the sets $B, TB, T^2B, \\dots, T^{n-1}B$ are pairwise disjoint, and\n2. $\\mu(B \\cup TB \\cup \\cdots \\cup T^{n-1}B) \u003e 1 - \\varepsilon$.\n\n## Remarks\n\nThe sets $B, TB, \\dots, T^{n-1}B$ form a Rokhlin tower of height $n$ with base $B$, and $T$ moves each level to the next one.\nThe proof for ergodic $T$ starts from a set $F$ of small positive measure and slices the space by the first return time to $F$.\nEach column of height at least $n$ is cut into blocks of height $n$, and the leftover has measure at most about $n \\mu(F)$.\nExample at work: a tower gives a periodic approximation of $T$.\nDefine $P = T$ on $B \\cup \\cdots \\cup T^{n-2}B$, let $P = T^{-(n-1)}$ on $T^{n-1}B$, and let $P$ be the identity elsewhere.\nThen $P^n$ is the identity, $P$ preserves $\\mu$, and $P$ agrees with $T$ outside a set of measure less than $\\mu(B) + \\varepsilon \\le 1/n + \\varepsilon$.\nThe aperiodicity hypothesis is necessary: if $T^2 = \\mathrm{id}$ and $n = 3$, then $T^2 B = B$, so condition 1 forces $\\mu(B) = 0$ and condition 2 fails.\nThe error $\\varepsilon$ cannot be removed in general.\nIf a tower of height $n \\ge 2$ covers $X$ up to a null set, then $T^n B = B$ up to a null set, and $f = \\sum_{j=0}^{n-1} e^{2\\pi i j/n} \\mathbf{1}_{T^j B}$ satisfies $f \\circ T = e^{2\\pi i/n} f$.\nSo $f$ is a nonconstant eigenfunction, which is impossible when $T$ is weakly mixing, for example a Bernoulli scheme.\n\n## Explanations\n\n### L1 human\n\nAny measure-preserving map without periodic behaviour can be pictured, up to a tiny leftover, as a tall stack of $n$ floors.\nThe map moves each floor up to the next one, and only at the top floor does something complicated happen.\nThe height $n$ can be as large as you like and the leftover as small as you like.\n\n### L2 human\n\nThe lemma is the main tool for building approximations and codings in ergodic theory.\nIt shows that every aperiodic transformation can be approximated in the weak topology by periodic ones.\nIt is used in the proofs of Sinai's factor theorem and of Ornstein's isomorphism theorem.\nIt extends to free actions of $\\Z^d$ and of countable amenable groups, where the role of intervals of times is played by Følner sets.\n",
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