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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\mu)$ be a measure space.\nLet $f_1, f_2, \\dots \\colon X \\to [0, \\infty]$ be [[measurable functions|YYNCJSR0PJ]] with $f_1(x) \\le f_2(x) \\le \\cdots$ for every $x \\in X$.\nLet $f(x) = \\lim_{n \\to \\infty} f_n(x)$, a limit in $[0, \\infty]$.\nThen $f$ is measurable and\n$$\\int_X f \\, d\\mu = \\lim_{n \\to \\infty} \\int_X f_n \\, d\\mu,$$\nwhere the integrals are [[integrals of non-negative functions|DG76G5Z6XW]].\n\n## Remarks\n\nProof idea: the integrals $\\int f_n$ increase and are at most $\\int f$, so their limit $L$ satisfies $L \\le \\int f$.\nFor the reverse bound, fix a simple $s \\le f$ and $c \\in (0, 1)$.\nThe sets $E_n = \\set{f_n \\ge c s}$ increase to $X$, so $\\int f_n \\ge c \\int_{E_n} s \\to c \\int s$ by continuity of measure from below.\nLetting $c \\to 1$ and taking the supremum over $s$ gives $L \\ge \\int f$.\n\nExample: with $f_n = \\min(f, n)$ the theorem computes $\\int f$ from bounded functions.\nFor a series of non-negative measurable functions it gives $\\int \\sum_k g_k \\, d\\mu = \\sum_k \\int g_k \\, d\\mu$.\n\nBoth hypotheses matter.\nWithout monotonicity, $f_n = \\mathbf{1}_{[n, n+1]}$ on $\\R$ tends to $0$ pointwise while each integral is $1$.\nWithout non-negativity, $f_n = -\\mathbf{1}_{[n, \\infty)}$ increases to $0$ while each integral is $-\\infty$.\nIt holds also when $f_n \\le f_{n+1}$ only almost everywhere.\nThe analogous fact for sequences of numbers is the [[monotone convergence theorem for sequences|0FM6N4R0VB]].\n\n## Explanations\n\n### L1 human\n\nIf a sequence of non-negative functions grows at every point, then the integrals grow to the integral of the limit.\nNo extra bound on the functions is needed.\nThe value $\\infty$ is allowed on both sides.\n\n### L2 human\n\nThis is the first of the three convergence theorems of Lebesgue integration.\nFatou's lemma follows from it, and the dominated convergence theorem follows from Fatou's lemma.\nIt is also the tool to prove that the integral is additive, by approximating with increasing simple functions.\nIn probability it gives $\\mathbb{E}[\\sum_k Y_k] = \\sum_k \\mathbb{E}[Y_k]$ for non-negative random variables.\n",
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