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.
Assume is an interval and that for every natural number k, is an increasing function. If,
exists for all then,
almost everywhere in I.
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.
- ^ a b 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.