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
[edit]Assume is an interval and that for every natural number k, is an increasing function. If,
exists for all then for almost any the derivatives exist and are related as:[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]