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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a measure space with $\\mu$ a [[finite measure|63Z1Q4FS23]], and let $K$ be a subset of [[$L^1(\\mu)$|SASP14MA1Z]].\nThen the following are equivalent.\n\n1. $K$ is relatively compact in the [[weak topology|P6KHVSTPZV]] of $L^1(\\mu)$.\n2. $K$ is [[uniformly integrable|VRVVKMXDGV]]: $\\lim_{M \\to \\infty} \\sup_{f \\in K} \\int_{\\set{\\abs{f} \u003e M}} \\abs{f} \\, d\\mu = 0$.\n\n## Remarks\n\nWhen $\\mu$ is finite, condition 2 implies that $K$ is bounded in $L^1$.\nOne proof of 2 $\\Rightarrow$ 1 views $K$ inside the bidual $L^\\infty(\\mu)^*$ and takes a weak-$*$ cluster point.\nUniform integrability makes the set functions $E \\mapsto \\int_E f \\, d\\mu$, $f \\in K$, uniformly absolutely continuous, so the cluster point is a countably additive measure absolutely continuous with respect to $\\mu$.\nThe [[Radon–Nikodym theorem|1ZSZS5G1Q4]] then gives it as an element of $L^1(\\mu)$.\nThe converse uses the Eberlein–Šmulian theorem to reduce to sequences.\n\nExample: on $[0, 1]$ with Lebesgue measure, a set bounded in $L^2$ is uniformly integrable, because $\\int_{\\set{\\abs{f} \u003e M}} \\abs{f} \\le \\norm{f}_2^2 / M$.\nSo bounded subsets of $L^2$ are weakly relatively compact in $L^1$.\n\nBoundedness in $L^1$ alone is not enough.\nThe functions $f_n = n \\mathbf{1}_{[0, 1/n]}$ on $[0, 1]$ all have norm $1$, but $\\int_{\\set{f_n \u003e M}} f_n = 1$ for $n \u003e M$.\nNo subsequence converges weakly in $L^1$: as measures, $f_n \\, dx$ concentrate at the point $0$.\n\n## Explanations\n\n### L1 human\n\nIn $L^1$, bounded sequences need not have limits even in a weak sense, because the mass of the functions can pile up on smaller and smaller sets.\nThe theorem says that this concentration is the only obstruction.\nIf the functions cannot put a fixed amount of mass where they are very large, then every sequence has a weakly convergent subsequence.\n\n### L2 human\n\nThe theorem is the $L^1$ substitute for the weak compactness of bounded sets in reflexive spaces such as $L^p$, $1 \u003c p \u003c \\infty$.\nIn probability it is the analytic side of the fact that uniformly integrable families of random variables behave well under limits, as in the [[Vitali convergence theorem|C3CKZAZ57Z]].\nIn the calculus of variations it gives weak limits for sequences with superlinear integral bounds, by the de la Vallée-Poussin criterion.\n",
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