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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a measure space.\nLet $f_1, f_2, \\dots$ and $f$ be real or complex [[integrable functions|X7GTF3D98F]] on $X$ with $f_n \\to f$ [[almost everywhere|RARTMA0ZEC]].\nThen\n$$\\lim_{n \\to \\infty} \\int_X \\abs{f_n - f} \\, d\\mu = 0 \\quad \\text{if and only if} \\quad \\lim_{n \\to \\infty} \\int_X \\abs{f_n} \\, d\\mu = \\int_X \\abs{f} \\, d\\mu.$$\n\n## Remarks\n\nProof idea: the direct implication follows from $\\abs{\\int \\abs{f_n} - \\int \\abs{f}} \\le \\int \\abs{f_n - f}$.\nFor the converse, the functions $g_n = \\abs{f_n} + \\abs{f} - \\abs{f_n - f}$ are non-negative and tend to $2\\abs{f}$ almost everywhere.\n[[Fatou's lemma|S95TR54TCG]] gives $2 \\int \\abs{f} \\le 2 \\int \\abs{f} - \\limsup_n \\int \\abs{f_n - f}$, and $\\int \\abs{f}$ is finite.\n\nExample: let $p_n$ and $p$ be probability densities on $\\R$ with $p_n \\to p$ almost everywhere.\nThen $\\int p_n = \\int p = 1$, so $\\int \\abs{p_n - p} \\to 0$.\nThe distributions with densities $p_n$ then converge to the one with density $p$ in total variation.\n\nThe convergence of the norms is needed.\nFor $f_n = n \\mathbf{1}_{(0, 1/n)}$ on $\\R$, $f_n \\to 0$ everywhere but $\\int \\abs{f_n - 0} = 1$.\n\n## Explanations\n\n### L1 human\n\nSuppose functions converge at almost every point, and their total sizes converge to the total size of the limit.\nThen the area between their graphs and the graph of the limit goes to zero.\nNo mass can escape, because the total mass is kept.\n\n### L2 human\n\nScheffé's lemma is a quick criterion for convergence in $L^1$.\nIts main use is in probability and statistics: pointwise convergence of densities implies convergence of the distributions in total variation, hence in distribution.\nIt is close in spirit to the Vitali convergence theorem, but it holds on any measure space and its proof is a direct application of Fatou's lemma.\n",
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