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]

gollark: Anyone made a brainf\*\*k interpreter in brain\*uc\*?

gollark: But yes, probably.

gollark: There are 8 commands, I do not understand what you'd want to modify.

gollark: I actually charge my phone using 2500 potatoes.

gollark: Failing that, in relation to the osmarks.tk central server.

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.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[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.