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 [math]\displaystyle{ I \subseteq \mathbb R }[/math] is an interval and that for every natural number k, [math]\displaystyle{ f_k: I \to \mathbb R }[/math] is an increasing function. If,

[math]\displaystyle{ s(x) := \sum_{k=1}^\infty f_k(x) }[/math]

exists for all [math]\displaystyle{ x \in I, }[/math] then for almost any [math]\displaystyle{ x \in I, }[/math] the derivatives exist and are related as:

[math]\displaystyle{ s'(x) = \sum_{k=1}^\infty f_k'(x). }[/math]

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 [math]\displaystyle{ \sum_{k=1}^n f_k'(x) }[/math] on I for every n.