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Fubini's theorem on differentiation

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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 {\displaystyle I\subseteq \mathbb {R} } is an interval and that for every natural number k, f k : I → R {\displaystyle f_{k}:I\to \mathbb {R} } is an increasing function. If,

s ( x ) := ∑ k = 1 ∞ f k ( x ) {\displaystyle s(x):=\sum _{k=1}^{\infty }f_{k}(x)}

exists for all x ∈ I , {\displaystyle x\in I,} then for almost any x ∈ I , {\displaystyle x\in I,} the derivatives exist and are related as:

s ′ ( x ) = ∑ k = 1 ∞ f k ′ ( x ) . {\displaystyle s'(x)=\sum _{k=1}^{\infty }f_{k}'(x).}

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 ∑ k = 1 n f k ′ ( x ) {\displaystyle \sum _{k=1}^{n}f_{k}'(x)} on I for every n.

References

  1. ^ 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.
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