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  "body": "\n## Definition\n\nLet $(X, \\Sigma, \\mu)$ be a [[finite measure|63Z1Q4FS23]] space.\nLet $\\mathcal{F}$ be a family of real or complex [[integrable functions|X7GTF3D98F]] on $X$.\nThe family $\\mathcal{F}$ is *uniformly integrable* if\n$$\\lim_{M \\to \\infty} \\ \\sup_{f \\in \\mathcal{F}} \\int_{\\set{\\abs{f} \u003e M}} \\abs{f} \\, d\\mu = 0,$$\nwhere the integrals are [[integrals of non-negative functions|DG76G5Z6XW]].\n\n## Remarks\n\nExample: on $[0, 1]$ with Lebesgue measure, let $\\mathcal{F} = \\set{f : \\abs{f} \\le 2}$.\nFor $M \\ge 2$ each set $\\set{\\abs{f} \u003e M}$ is empty, so the supremum is $0$.\n\nNon-example: on $[0, 1]$, let $f_n = n \\mathbf{1}_{(0, 1/n)}$.\nFor $n \u003e M$ the integral of $f_n$ over $\\set{f_n \u003e M}$ equals $1$, so the supremum is $1$ for every $M$.\n\nOn a finite measure space the condition is equivalent to two conditions together.\nFirst, $\\sup_{f \\in \\mathcal{F}} \\int \\abs{f} \\, d\\mu \u003c \\infty$.\nSecond, for every $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ with $\\int_A \\abs{f} \\, d\\mu \u003c \\varepsilon$ for all $f \\in \\mathcal{F}$ whenever $\\mu(A) \u003c \\delta$.\nAny family dominated by one integrable function is uniformly integrable.\nBounded families in $L^p$ for some $p \u003e 1$ are uniformly integrable.\nOn infinite measure spaces other versions of the definition are in use.\n",
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