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  "body": "\n## Statement\n\nLet $X$ be a [[compact|0TVJZYB6SH]] [[metric space|0VKD1TP7F3]], and let $T \\colon X \\to X$ be a [[continuous map|0X3WGX95FE]].\nThen there is a Borel [[probability measure|059TJN0NZ1]] $\\mu$ on $X$ that is [[$T$-invariant|B232EZ1FMB]], that is, $\\mu(T^{-1}(A)) = \\mu(A)$ for every Borel set $A \\subseteq X$.\n\n## Remarks\n\nFix $x \\in X$ and let $\\mu_n = \\frac{1}{n} \\sum_{k=0}^{n-1} \\delta_{T^k(x)}$, where $\\delta_y$ is the point mass at $y$.\nSince $X$ is compact, the [[Borel probability measures|DYR1QFNGNJ]] on $X$ form a compact set for [[weak convergence|MBW9JCGZC1]], for instance by [[Prokhorov's theorem|D6FMEQYA76]].\nSo some subsequence $\\mu_{n_j}$ converges weakly to a probability measure $\\mu$.\nFor continuous $g \\colon X \\to \\R$ one has $\\int g \\circ T \\, d\\mu_n - \\int g \\, d\\mu_n = \\frac{1}{n}\\left(g(T^n(x)) - g(x)\\right)$, which tends to $0$.\nPassing to the limit gives $\\int g \\circ T \\, d\\mu = \\int g \\, d\\mu$ for all continuous $g$, and this forces $T_* \\mu = \\mu$.\nThe same argument gives an invariant probability measure for a continuous flow on a compact metric space.\n\nCompactness is needed: the translation $T(x) = x + 1$ on $\\R$ has no invariant probability measure.\nIndeed, invariance would give $\\mu([k, k+1)) = \\mu([0, 1))$ for every $k \\in \\Z$, and these countably many disjoint sets cover $\\R$, so their measures cannot sum to $1$.\n\nExample: for the rotation $x \\mapsto x + \\alpha \\bmod 1$ of the circle, the theorem produces an invariant measure; when $\\alpha$ is irrational, it is Lebesgue measure, which is the only invariant one.\n\n## Explanations\n\n### L1 human\n\nAny continuous motion of a compact space has a way of spreading out mass that the motion leaves unchanged.\nOne finds it by following one orbit for a long time and recording how often it visits each region.\nThese visit frequencies settle down, at least along a subsequence, to an invariant measure.\n\n### L2 human\n\nThe theorem makes ergodic theory available for every continuous map of a compact metric space: there is always at least one invariant probability measure to work with.\nThe set of invariant probability measures is a nonempty compact convex set, and its extreme points are the ergodic measures, so ergodic invariant measures exist as well.\nWhen the invariant measure is unique, the map is called uniquely ergodic, and then Birkhoff averages of continuous functions converge uniformly.\nThe variational principle for topological entropy is a supremum over this set of invariant measures.\n",
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