Fubini's theorem on differentiation

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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]


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]


  1. ^ a b 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.