Fubini's theorem on differentiation - Alchetron, the free social encyclopedia

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.

Statement

Assume I ⊆ R is an interval and that for every natural number k, f k : I → R is an increasing function. If,

s ( x ) := ∑ k = 1 ∞ f k ( x )

exists for all x ∈ I , then,

s ′ ( x ) = ∑ k = 1 ∞ f k ′ ( x )

almost everywhere in I.

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 ∑ k = 1 n f k ′ ( x ) on I for every n.

References

Fubini's theorem on differentiation Wikipedia

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