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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] and let $\\mu$ be a [[Borel measure|DYR1QFNGNJ]] on $X$.\nFor $x \\in X$ and $r \u003e 0$, let $B(x, r)$ be the [[open ball|0YE5868HZB]] of centre $x$ and radius $r$.\nThe measure $\\mu$ is *doubling* if:\n\n1. $0 \u003c \\mu(B(x, r)) \u003c \\infty$ for all $x \\in X$ and $r \u003e 0$.\n2. There is a constant $C \\ge 1$ with $\\mu(B(x, 2r)) \\le C \\, \\mu(B(x, r))$ for all $x \\in X$ and $r \u003e 0$.\n\n## Remarks\n\nExample: Lebesgue measure $\\lambda$ on $\\R^n$ is doubling with $C = 2^n$.\nIndeed $\\lambda(B(x, r)) = \\omega_n r^n$ with $\\omega_n$ the volume of the unit ball, so the ratio is exactly $2^n$.\n\nNon-example: the standard Gaussian measure $\\gamma$ on $\\R$ satisfies clause 1 but not clause 2.\nLet $\\varphi(t) = e^{-t^2/2}/\\sqrt{2\\pi}$ be its density, which decreases on $[0, \\infty)$, and let $x \u003e 2$.\nThen $\\gamma(B(x, 1)) \\le 2 \\varphi(x - 1)$ and $\\gamma(B(x, 2)) \\ge \\frac12 \\varphi(x - \\frac32)$.\nSo $\\gamma(B(x,2)) / \\gamma(B(x,1)) \\ge \\frac14 e^{(x - 5/4)/2}$, which is unbounded as $x \\to \\infty$.\n\nSome authors drop clause 1 and only ask for clause 2 for balls of positive finite measure.\nA metric space with a doubling measure is a doubling metric space: each ball is covered by a bounded number of balls of half the radius.\n",
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