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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 *weakly mixing* if for all $A, B \\in \\mathcal{B}$,\n$$\\lim_{N \\to \\infty} \\frac{1}{N} \\sum_{n=0}^{N-1} \\abs{\\mu(T^{-n} A \\cap B) - \\mu(A) \\, \\mu(B)} = 0.$$\n\n## Remarks\n\nExample: every [[mixing|S3RJQ9CX2P]] map is weakly mixing, since the averages of a sequence that tends to $0$ also tend to $0$.\nSo the doubling map $x \\mapsto 2x \\bmod 1$ with Lebesgue measure is weakly mixing.\n\nNon-example: let $T(x) = x + \\alpha \\bmod 1$ with $\\alpha$ irrational, and let $f(x) = e^{2\\pi i x}$.\nThen $f \\circ T = e^{2\\pi i \\alpha} f$, so $f$ is a nonconstant eigenfunction of the [[Koopman operator|37TRG5DBQC]].\nBy form 2 below, $T$ is not weakly mixing, although it is ergodic.\n\nEquivalent forms, for $T$ as above:\n\n1. The product map $T \\times T$ is ergodic for the product measure $\\mu \\times \\mu$.\n2. Every $f \\in L^2(\\mu)$ with $f \\circ T = \\lambda f$ for some $\\lambda \\in \\C$ is constant almost everywhere.\n3. For all $A, B \\in \\mathcal{B}$ there is a set $J \\subseteq \\N$ of density zero such that $\\mu(T^{-n} A \\cap B) \\to \\mu(A)\\mu(B)$ as $n \\to \\infty$ with $n \\notin J$.\n\nThere are weakly mixing maps that are not mixing; Chacon's transformation is a classical example.\n",
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