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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 measurable functions on $X$ with $f_n \\to f$ [[almost everywhere|RARTMA0ZEC]].\nSuppose there is an [[integrable function|X7GTF3D98F]] $g \\colon X \\to [0, \\infty]$ with $\\abs{f_n} \\le g$ almost everywhere for every $n$.\nThen $f$ and each $f_n$ are integrable, and\n$$\\lim_{n \\to \\infty} \\int_X \\abs{f_n - f} \\, d\\mu = 0, \\qquad \\lim_{n \\to \\infty} \\int_X f_n \\, d\\mu = \\int_X f \\, d\\mu,$$\nwhere the integrals are [[Lebesgue integrals|DWJF7ABPR1]].\n\n## Remarks\n\nProof idea: $\\abs{f} \\le g$ almost everywhere, so $f$ is integrable.\nThe functions $2g - \\abs{f_n - f}$ are non-negative almost everywhere and tend to $2g$.\n[[Fatou's lemma|S95TR54TCG]] gives $\\int 2g \\le \\int 2g - \\limsup_n \\int \\abs{f_n - f}$, so $\\limsup_n \\int \\abs{f_n - f} = 0$.\nThe second limit follows from $\\abs{\\int f_n - \\int f} \\le \\int \\abs{f_n - f}$.\n\nExample: $\\int_0^1 x^n \\, dx \\to 0$, since $x^n \\to 0$ for $x \\in [0, 1)$ and $\\abs{x^n} \\le 1$, which is integrable on $[0, 1]$.\n\nThe dominating function is needed.\nFor $f_n = n \\mathbf{1}_{(0, 1/n)}$ on $\\R$, $f_n \\to 0$ everywhere but $\\int f_n = 1$.\nHere $\\sup_n f_n$ behaves like $1/x$ near $0$ and is not integrable.\n\n## Explanations\n\n### L1 human\n\nSuppose a sequence of functions converges at almost every point, and all of them stay below one fixed function of finite integral.\nThen their integrals converge to the integral of the limit.\nThe fixed bound stops mass from escaping to infinity or piling up in a spike.\n\n### L2 human\n\nThis is the most used convergence theorem of analysis.\nIt justifies differentiating under the integral sign, exchanging sums and integrals, and passing to the limit in many approximation arguments.\nFor finite measure spaces, a constant bound is enough, which gives the bounded convergence theorem.\nThe Vitali convergence theorem replaces domination by the weaker uniform integrability and gives a necessary and sufficient condition.\n",
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