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  "body": "\n## Definition\n\nLet $(X, \\mathcal{B}, \\mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \\colon X \\to X$ be a [[measure-preserving transformation|556WBDAHNP]].\nThe map $T$ is *mixing*, or *strongly mixing*, if for all $A, B \\in \\mathcal{B}$,\n$$\\lim_{n \\to \\infty} \\mu(T^{-n} A \\cap B) = \\mu(A) \\, \\mu(B).$$\n\n## Remarks\n\nExample: let $T(x) = 2x \\bmod 1$ on $[0,1)$ with Lebesgue measure $\\mu$.\nLet $A$ and $B$ be dyadic intervals of length $2^{-k}$, that is, of the form $[j 2^{-k}, (j+1) 2^{-k})$.\nFor $n \\ge k$, the set $T^{-n} A$ is a union of $2^n$ intervals of length $2^{-n-k}$, one in each interval $[i 2^{-n}, (i+1) 2^{-n})$.\nExactly $2^{n-k}$ of them lie in $B$, so $\\mu(T^{-n} A \\cap B) = 2^{-2k} = \\mu(A) \\mu(B)$.\nDyadic intervals generate the Borel sets, and an approximation argument gives the limit for all $A$ and $B$.\n\nNon-example: let $T(x) = x + \\alpha \\bmod 1$ with $\\alpha$ irrational, and let $A = B = [0, 1/4)$.\nThe numbers $n \\alpha \\bmod 1$ come arbitrarily close to $0$ for infinitely many $n$, and for those $n$ the measure $\\mu(T^{-n} A \\cap A)$ is close to $1/4$.\nSo it does not tend to $1/16$, and the rotation is not mixing, although it is ergodic.\n\nTaking $A = B$ with $T^{-1} A = A$ shows that a mixing map is ergodic: then $\\mu(A) = \\mu(A)^2$, so $\\mu(A) \\in \\set{0, 1}$.\nIn terms of functions, $T$ is mixing if and only if $\\int (f \\circ T^n) \\, g \\, d\\mu \\to \\int f \\, d\\mu \\int g \\, d\\mu$ for all $f, g \\in L^2(\\mu)$.\n",
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