Suppose that $(f_n)$ is a sequence of monotone non-decreasing functions on $[a,b]$ such that $f(x) = \sum_{n=1} ^\infty f_n (x)$ is finite for each $x\in [a,b]$. By Lebesgue's theorem on monotone functions, each $f_n$ is differentiable almost everywhere, and it is clear that $f$ is monotone as well, so $f$ is also differentiable a.e.
Must it be the case that \sum_{n=1} ^\infty f_n' = f' almost everywhere? I know that for each $N$ and for each $h \in \mathbb R$, $ \frac{1}{h}\sum_{n=1} ^N [f_n(x+h) - f(x)] \leq \frac{1}{h}\sum_{n=1} ^\infty [f_n(x+h) - f(x)] $ and taking the limit as $h \rightarrow 0$ gives \sum_{n=1} ^N f_n '\leq f', and so \sum_{n=1} ^\infty f_n ' \leq f'. Is the reverse inequality also true?