Exercise 13, § 6

In a compact interval $I = [a, b]$ of $\mathbf{R}$, let $(f_n)$ be a sequence of continuous, increasing finite functions, such that $f_n(a) = 0$ for all $n$ and such that the sum $s(x) = \sum_{n=1}^\infty f_n(x)$ is finite on $I$. Show that there exists a set $N \subset I$, negligible for Lebesgue measure, such that, for every $x \in I - N$, the derivatives $f'_n(x)$ and $s'(x)$ exist and $s'(x) = \sum_{n=1}^\infty f'_n(x)$ ('Fubini's theorem'). (Use Exer. 12. Observe that the series with general term $f'_n(x)$ is convergent almost everywhere; setting

consider next a subsequence $(s_{n_k})$ such that the series with general term $s(x) - s_{n_k}(x)$ is convergent on $I$, and apply the preceding remark to this series.)