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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a [[finite measure|63Z1Q4FS23]] space.\nLet $f_1, f_2, \\dots$ be real or complex [[integrable functions|X7GTF3D98F]] on $X$.\nLet $f$ be a measurable function with $f_n \\to f$ [[almost everywhere|RARTMA0ZEC]].\nThen the following are equivalent.\n\n1. The family $\\set{f_n : n \\ge 1}$ is [[uniformly integrable|VRVVKMXDGV]].\n2. The function $f$ is integrable and $\\lim_{n \\to \\infty} \\int_X \\abs{f_n - f} \\, d\\mu = 0$.\n\n## Remarks\n\nProof idea for 1 implies 2: uniform integrability bounds $\\int \\abs{f_n}$, so $f$ is integrable by [[Fatou's lemma|S95TR54TCG]].\nSplit $\\int \\abs{f_n - f}$ into the part on a set where $f_n \\to f$ uniformly, given by Egorov's theorem, and the part on a small set.\nUniform integrability makes the second part small, uniformly in $n$.\nFor 2 implies 1, a sequence converging in $L^1$ is uniformly integrable, because finitely many integrable functions are, and the tail is close to $f$.\n\nExample: if $\\sup_n \\int \\abs{f_n}^2 \\, d\\mu \u003c \\infty$ and $f_n \\to f$ almost everywhere, then the family is uniformly integrable and $f_n \\to f$ in $L^1$.\n\nThe finiteness of $\\mu$ is needed in this form.\nOn $\\R$ with Lebesgue measure, $f_n = \\frac{1}{n} \\mathbf{1}_{[0, n]}$ satisfies the condition of uniform integrability, since $\\abs{f_n} \\le 1$, and tends to $0$ everywhere, but $\\int \\abs{f_n} = 1$ for all $n$.\nThe theorem also holds with convergence in measure in place of convergence almost everywhere.\n\n## Explanations\n\n### L1 human\n\nPointwise convergence alone does not make integrals converge, because mass can escape into thin tall spikes.\nOn a space of finite total measure, uniform integrability is exactly the condition that forbids such spikes.\nSo it is the precise condition for convergence of the integrals.\n\n### L2 human\n\nThe theorem refines the dominated convergence theorem: a dominated sequence is uniformly integrable, but not conversely.\nIt is the form of the convergence theorem used in probability, where one shows uniform integrability with a moment bound such as $\\sup_n \\mathbb{E}[\\abs{Y_n}^p] \u003c \\infty$ for some $p \u003e 1$.\nThere are versions for $L^p$ and for infinite measure spaces, where a condition of tightness is added.\n",
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