Fubini's theorem on differentiation

In mathematics, Fubini's theorem on differentiation, named after Guido Fubini, is a result in real analysis concerning the differentiation of series of monotonic functions. It can be proven by using Fatou's lemma and the properties of null sets.

Statement

Assume $I\subseteq \mathbb {R}$ is an interval and that for every natural number k, $f_{k}:I\to \mathbb {R}$ is an increasing function. If,

$s(x):=\sum _{k=1}^{\infty }f_{k}(x)$

exists for all $x\in I,$ then for almost any $x\in I,$ the derivatives exist and are related as:

$s'(x)=\sum _{k=1}^{\infty }f_{k}'(x).$

In general, if we don't suppose f_k is increasing for every k, in order to get the same conclusion, we need a stricter condition like uniform convergence of $\sum _{k=1}^{n}f_{k}'(x)$ on I for every n.