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 $I \subseteq \mathbb R$ 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,

s'(x) = \sum_{k=1}^\infty f_k'(x)

almost everywhere in I.^[1]

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.^[2]

References

1 2 Jones, Frank (2001), Lebesgue Integration on Euclidean Space, Jones and Bartlett publishers, pp. 527–529.
↑ Rudin, Walter (1976), Principles of Mathematical Analysis, McGraw-Hill, p. 152.

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