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 is an interval and that for every natural number k, is an increasing function. If,
exists for all then,
almost everywhere in I.[1]
In general, if we don't suppose fk is increasing for every k, in order to get the same conclusion, we need a stricter condition like uniform convergence of on I for every n.[2]
References
This article is issued from
Wikipedia.
The text is licensed under Creative Commons - Attribution - Sharealike.
Additional terms may apply for the media files.