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 fk 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.