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.^[1]

Statement

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

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

exists for all ${\displaystyle x\in I,}$ then for almost any ${\displaystyle x\in I,}$ the derivatives exist and are related as:^[1]

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

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 ${\displaystyle \sum _{k=1}^{n}f_{k}'(x)}$ on I for every n.^[2]