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]

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References

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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[a-1] Jones, Frank (2001), Lebesgue Integration on Euclidean Space, Jones and Bartlett publishers, pp. 527–529.

[2] Rudin, Walter (1976), Principles of Mathematical Analysis, McGraw-Hill, p. 152.