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  "body": "\n## Statement\n\nLet $a \u003c b$ be real numbers and let $f_n \\colon [a, b] \\to \\R$, for $n \\ge 1$, be nondecreasing functions.\nAssume that the series $F(x) = \\sum_{n=1}^{\\infty} f_n(x)$ converges for every $x \\in [a, b]$.\nThen for [[almost every|RARTMA0ZEC]] $x \\in [a, b]$, with respect to [[Lebesgue measure|2Z2QVWZ262]], the [[derivatives|0F1WR9C83K]] $F'(x)$ and $f_n'(x)$ exist for all $n$, and\n$$F'(x) = \\sum_{n=1}^{\\infty} f_n'(x).$$\n\n## Remarks\n\nReplacing $f_n$ by $f_n - f_n(a)$ changes nothing, so one may assume $f_n \\ge 0$.\nBy [[Lebesgue's theorem|1Y4N1DDQNR]], $F$ and all $f_n$ are differentiable almost everywhere.\nSince $F$ minus a partial sum is nondecreasing, the partial sums of $\\sum f_n'$ are bounded by $F'$ almost everywhere, so the series of derivatives converges.\nChoosing partial sums $S_{n_k}$ with $F(b) - S_{n_k}(b) \u003c 2^{-k}$ and applying the same argument to $\\sum_k (F - S_{n_k})$ shows that the terms $F' - S_{n_k}'$ tend to $0$ almost everywhere.\nMonotonicity is essential: take $f_1(x) = \\sin x$ and $f_n(x) = \\sin(n^2 x)/n^2 - \\sin((n-1)^2 x)/(n-1)^2$ for $n \\ge 2$; the partial sums $\\sin(n^2 x)/n^2$ converge uniformly to $0$, but their derivatives $\\cos(n^2 x)$ do not converge to $0$.\n\n## Explanations\n\n### L1 human\n\nA convergent series of increasing functions can be differentiated term by term, except on a set of length zero.\nNo uniform convergence and no smoothness is needed, only that each term is increasing.\n\n### L2 human\n\nThe theorem is a tool in the theory of monotone functions and functions of [[bounded variation|A8251BHCMP]].\nIt shows, for example, that a jump function, a convergent sum of step functions, has derivative $0$ almost everywhere.\nIt contrasts with the classical termwise differentiation theorem, which asks for uniform convergence of the derivatives.\n",
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