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  "body": "\n## Statement\n\nLet $(X, \\mathcal{A}, \\mu)$ be a [[probability space|0NFPJVJJHX]], let $T \\colon X \\to X$ be a [[measure-preserving transformation|556WBDAHNP]], and let $f \\colon X \\to \\R$ be [[integrable|X7GTF3D98F]].\nFor $n \\ge 1$ let $S_n f = \\sum_{k=0}^{n-1} f \\circ T^k$ be the [[Birkhoff sums|8TD2KCVH5N]] of $f$, and let\n$$E = \\set{x \\in X : \\sup_{n \\ge 1} S_n f(x) \u003e 0}.$$\nThen\n$$\\int_E f \\, d\\mu \\ge 0.$$\n\n## Remarks\n\nProof idea, after Garsia: let $F_N = \\max(0, S_1 f, \\dots, S_N f)$.\nFor $1 \\le n \\le N$ one has $S_n f \\le f + F_N \\circ T$, so $F_N \\le f + F_N \\circ T$ on the set $E_N = \\set{F_N \u003e 0}$.\nIntegrating over $E_N$ and using $F_N \\ge 0$ and the invariance of $\\mu$ gives $\\int_{E_N} f \\, d\\mu \\ge \\int_X F_N \\, d\\mu - \\int_X F_N \\circ T \\, d\\mu = 0$.\nThe sets $E_N$ increase to $E$, and dominated convergence finishes the proof.\n\nApplied to $f - \\lambda$ for a real number $\\lambda \u003e 0$, it gives the maximal inequality\n$$\\lambda \\, \\mu\\left(\\set{x : \\sup_{n \\ge 1} \\tfrac{1}{n} S_n f(x) \u003e \\lambda}\\right) \\le \\int_X \\abs{f} \\, d\\mu.$$\nThe inequality $\\int_E f \\, d\\mu \\ge 0$ holds, with the same proof, for a measure-preserving map of any measure space and any integrable $f$.\n\nExample: if $f \\ge 0$ is not almost everywhere $0$, then $E = \\set{x : f(T^k(x)) \u003e 0 \\text{ for some } k \\ge 0}$, and the conclusion $\\int_E f \\, d\\mu \\ge 0$ is clear; the content of the theorem lies in functions of both signs.\n\n## Explanations\n\n### L1 human\n\nConsider the points from which the running total of $f$ along the orbit becomes positive at some time.\nThe theorem says that, on this set, the function $f$ has a nonnegative integral.\nIn other words, the points where the running sum ever gets ahead cannot, on average, be points where $f$ is negative.\n\n### L2 human\n\nThis is the key lemma in the standard proof of Birkhoff's pointwise ergodic theorem.\nThrough the maximal inequality, it controls the supremum of all Birkhoff averages at once.\nThis control is what is needed to pass from a dense class of functions to all integrable functions.\nIt plays the same role as the Hardy–Littlewood maximal inequality in the proof of the Lebesgue differentiation theorem.\n",
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