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  "body": "\n## Summary\n\nAn invariant measure of a map is a [[measure|PR11BZQQPG]] that the map does not change: every set has the same measure as its preimage.\nWith respect to such a measure the map is a [[measure-preserving transformation|556WBDAHNP]], and the tools of ergodic theory apply.\nInvariant probability measures describe the statistics of orbits, such as how often an orbit visits a region.\nOn a compact metric space every continuous map has at least one invariant probability measure, by the [[Krylov–Bogolyubov theorem|1DHZ81S0YE]].\n\n## Explanations\n\n### L1 human\n\nThink of a measure as a way of spreading sand over the space.\nApply the map to every grain: the measure is invariant if the sand ends up spread exactly as before.\nLebesgue measure on the circle is invariant under every rotation, since rotating does not change lengths of arcs.\nFor the map $x \\mapsto x/2$ on the real line, the point mass at $0$ is invariant, while Lebesgue measure is not, since the map squeezes everything towards $0$.\n",
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