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  "body": "\n## Statement\n\nLet $(X, \\mathcal{B}, \\mu)$ be a [[standard probability space|QDDRDW9PXS]] and let $T \\colon X \\to X$ be an invertible [[ergodic|AZRPHS478B]] [[measure-preserving transformation|556WBDAHNP]] with measurable inverse.\nAssume that its [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] $h_\\mu(T)$ is positive.\nLet $B$ be a two-sided [[Bernoulli scheme|D33JK3AXRZ]] on a finite alphabet, acting on $(Y, \\nu)$, with entropy $h(B) \\le h_\\mu(T)$.\nThen $B$ is a factor of $T$: there is a measurable map $\\varphi \\colon X \\to Y$ with $\\mu(\\varphi^{-1}C) = \\nu(C)$ for every measurable $C \\subseteq Y$ and $\\varphi \\circ T = B \\circ \\varphi$ almost everywhere.\n\n## Remarks\n\nA factor map can only lower entropy, so the hypothesis $h(B) \\le h_\\mu(T)$ is necessary.\nErgodicity is needed too.\nLet $T$ be the disjoint union, with weight $1/2$ each, of a Bernoulli scheme $B'$ and an identity map, so that $h_\\mu(T) = h(B')/2 \u003e 0$.\nLet $B$ be any Bernoulli scheme with at least two symbols of positive weight.\nA factor map $\\varphi$ onto $B$ satisfies $\\varphi(x) = B(\\varphi(x))$ on the identity part.\nSo it sends a set of measure $1/2$ into the fixed points of $B$, which form a null set.\nSo $T$ has no such factor, whatever the entropy of $B$.\nThe proof constructs the factor map as a limit of finite codings, using the Rokhlin lemma to build towers and the Shannon–McMillan–Breiman theorem to count typical names.\nExample: an irrational rotation has entropy $0$, so the theorem says nothing about it.\nIndeed it has no nontrivial Bernoulli factor, since every factor of a zero-entropy system has entropy $0$.\nExample: every ergodic toral automorphism of positive entropy has Bernoulli factors of every entropy up to its own.\n\n## Explanations\n\n### L1 human\n\nAny ergodic system with some randomness, measured by positive entropy, contains a perfectly random process inside it.\nMore precisely, by watching the system through a suitable lens one sees a sequence of independent die rolls.\nThe die can be any loaded die whose entropy does not exceed the entropy of the system.\nSo positive entropy always means genuine randomness of the coin-tossing kind.\n\n### L2 human\n\nIt was a step toward the Ornstein isomorphism theorem, whose proof sharpens the same kind of coding argument from factor maps to isomorphisms.\nIt shows that positive entropy, a number defined by counting, forces a concrete structure: an independent process sitting inside the system.\n",
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