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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.\nLet $M(X, T)$ be the set of $T$-[[invariant|B232EZ1FMB]] [[Borel|DYR1QFNGNJ]] [[probability measures|059TJN0NZ1]] on $X$, with the topology of [[weak convergence of measures|MBW9JCGZC1]].\nLet $E(X, T) \\subseteq M(X, T)$ be the subset of measures for which $T$ is [[ergodic|AZRPHS478B]].\n\n1. $M(X, T)$ is a convex compact set, and $E(X, T)$ is exactly the set of its [[extreme points|3MSGFJM2NX]].\n2. For every $\\mu \\in M(X, T)$ there is a unique Borel probability measure $\\tau$ on $M(X, T)$ with $\\tau(E(X, T)) = 1$ and\n$$\\mu(A) = \\int_{E(X, T)} \\nu(A) \\, d\\tau(\\nu) \\quad \\text{for every Borel set } A \\subseteq X.$$\n\n## Remarks\n\nFor part 1, suppose $\\mu$ is not ergodic, and let $A$ be an invariant set with $0 \u003c \\mu(A) \u003c 1$.\nThen $\\mu$ is the convex combination $\\mu(A) \\mu_A + (1 - \\mu(A)) \\mu_{X \\setminus A}$ of the normalized restrictions, which are distinct invariant measures, so $\\mu$ is not extreme.\nConversely, if $\\mu$ is ergodic and $\\mu = t \\nu_1 + (1-t) \\nu_2$ with $0 \u003c t \u003c 1$, then $\\nu_1$ is absolutely continuous with respect to $\\mu$ with an invariant density, which ergodicity forces to be constant, so $\\nu_1 = \\mu$.\nPart 2 follows from Choquet theory, since $M(X, T)$ is a compact convex metrizable set whose barycentric decompositions are unique.\nThe decomposition can also be built from the conditional measures of $\\mu$ on the σ-algebra of invariant sets.\nExample: for the identity map, every Borel probability measure is invariant, and the ergodic ones are the point masses $\\delta_x$.\nThe decomposition of $\\mu$ is $\\mu = \\int \\delta_x \\, d\\mu(x)$, that is, $\\tau$ is the image of $\\mu$ under $x \\mapsto \\delta_x$.\nExample: for the rotation of the circle by a quarter turn, Lebesgue measure is the average over $x \\in [0, 1/4)$ of the uniform measures on the four-point orbits of $x$.\n\n## Explanations\n\n### L1 human\n\nEvery invariant measure can be split, in a unique way, into ergodic pieces.\nEach piece is a measure that cannot be split further, and the original measure is a weighted average of the pieces.\nSo to understand all invariant measures, it is enough to understand the ergodic ones.\n\n### L2 human\n\nMany theorems are first proved for ergodic systems, where time averages are constant, and then extended to all systems by the ergodic decomposition.\nFor example, [[Birkhoff's ergodic theorem|J6NMBX3E55]] for a non-ergodic measure gives a limit that is constant on each ergodic component.\nTwo distinct ergodic measures are mutually singular.\nBy part 1, the unique invariant measure of a [[uniquely ergodic|NZZE9KCAD8]] map is ergodic.\nThat $M(X, T)$ is not empty is the [[Krylov–Bogolyubov theorem|1DHZ81S0YE]].\n",
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