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.

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,

exists for all ${\displaystyle x\in I,}$ then,

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