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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]].\nLet $A_n f = \\frac{1}{n} \\sum_{k=0}^{n-1} f \\circ T^k$ be the [[Birkhoff averages|8TD2KCVH5N]] of $f$, and let $\\mathcal{I} = \\set{A \\in \\mathcal{A} : T^{-1}(A) = A}$ be the $\\sigma$-algebra of invariant sets.\nThen:\n\n1. $A_n f(x)$ converges as $n \\to \\infty$ for [[almost every|RARTMA0ZEC]] $x \\in X$, to the [[conditional expectation|0TTDGTEFMN]] $f^* = \\mathbb{E}(f \\mid \\mathcal{I})$;\n2. the limit satisfies $f^* \\circ T = f^*$ almost everywhere and $\\int_X f^* \\, d\\mu = \\int_X f \\, d\\mu$;\n3. $A_n f \\to f^*$ also in $L^1(\\mu)$;\n4. if $T$ is [[ergodic|AZRPHS478B]], then $f^* = \\int_X f \\, d\\mu$ almost everywhere.\n\n## Remarks\n\nThe usual proof combines the [[maximal ergodic theorem|V7PA1V3N0K]] with a density argument.\nFor functions of the form $h + g - g \\circ T$, with $h$ invariant and $g$ bounded, the averages converge everywhere, because the sums of $g - g \\circ T$ telescope.\nThe maximal inequality bounds $\\sup_n \\abs{A_n f}$ in measure by $\\norm{f}_1$, and this lets convergence pass to all integrable $f$.\nFor $f \\in L^2(\\mu)$, convergence in $L^2$ is the content of the [[mean ergodic theorem|PQPDQCCD02]].\n\nExample: let $T(x) = 2x \\bmod 1$ on $[0, 1)$ with Lebesgue measure, which is ergodic, and let $f$ be the indicator of $[1/2, 1)$.\nThen $f(T^k(x))$ is the $(k+1)$-st binary digit of $x$, so the theorem says that almost every $x$ has, in the limit, half of its binary digits equal to $1$.\n\nApplied to a sequence of independent identically distributed random variables, viewed as a shift on a product space, the theorem gives the [[strong law of large numbers|0DFWHQ839T]].\n\nThe measure must be invariant: for $T(x) = x/2$ on $[0, 1]$ with Lebesgue measure, which is not $T$-invariant, and $f(x) = x$, the averages tend to $0$ everywhere, while $\\int_0^1 f \\, dx = 1/2$, so conclusion 2 fails.\n\n## Explanations\n\n### L1 human\n\nFollow one orbit for a long time and average a quantity along it: this is a time average.\nAverage the same quantity over the whole space: this is a space average.\nFor an ergodic system the two agree for almost every starting point.\nSo watching one typical trajectory long enough reveals the statistics of the whole system.\n\n### L2 human\n\nThe theorem is the basic convergence result of ergodic theory.\nWithout ergodicity, the limit is still defined almost everywhere, but it can differ on different invariant pieces, and it equals the conditional expectation on the invariant sets.\nIt is used to prove the strong law of large numbers, the normality of almost every real number, and statistical properties of billiards and geodesic flows.\nThe same statement holds for measure-preserving flows, with sums replaced by integrals over time.\nKingman's subadditive ergodic theorem extends it from sums $S_n f$ to subadditive sequences, and the ergodic decomposition reduces the non-ergodic case to the ergodic one.\n",
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