Math 2.0

Real functions Monotone functions, bounded variation, absolute continuity Theorem

Fubini's theorem on differentiation

Builds on Monotone functionDerivative

Statement

Let a<ba < b be real numbers and let fn ⁣:[a,b]→ℝf_n \colon [a, b] \to \R, for n≥1n \ge 1, be nondecreasing functions. Assume that the series F(x)=∑n=1∞fn(x)F(x) = \sum_{n=1}^{\infty} f_n(x) converges for every x∈[a,b]x \in [a, b]. Then for almost every x∈[a,b]x \in [a, b], with respect to Lebesgue measure, the derivatives F′(x)F'(x) and fn′(x)f_n'(x) exist for all nn, and F′(x)=∑n=1∞fn′(x).F'(x) = \sum_{n=1}^{\infty} f_n'(x).

Remarks

Replacing fnf_n by fn−fn(a)f_n - f_n(a) changes nothing, so one may assume fn≥0f_n \ge 0. By Lebesgue's theorem, FF and all fnf_n are differentiable almost everywhere. Since FF minus a partial sum is nondecreasing, the partial sums of ∑fn′\sum f_n' are bounded by F′F' almost everywhere, so the series of derivatives converges. Choosing partial sums SnkS_{n_k} with F(b)−Snk(b)<2−kF(b) - S_{n_k}(b) < 2^{-k} and applying the same argument to ∑k(F−Snk)\sum_k (F - S_{n_k}) shows that the terms F′−Snk′F' - S_{n_k}' tend to 00 almost everywhere. Monotonicity is essential: take f1(x)=sin⁡xf_1(x) = \sin x and fn(x)=sin⁡(n2x)/n2−sin⁡((n−1)2x)/(n−1)2f_n(x) = \sin(n^2 x)/n^2 - \sin((n-1)^2 x)/(n-1)^2 for n≥2n \ge 2; the partial sums sin⁡(n2x)/n2\sin(n^2 x)/n^2 converge uniformly to 00, but their derivatives cos⁡(n2x)\cos(n^2 x) do not converge to 00.

Going deeper

For a student

The theorem is a tool in the theory of monotone functions and functions of bounded variation. It shows, for example, that a jump function, a convergent sum of step functions, has derivative 00 almost everywhere. It contrasts with the classical termwise differentiation theorem, which asks for uniform convergence of the derivatives.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionLimit point of a set, definition, no evidence levelLimit point of a setdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionDerivative, definition, no evidence levelDerivativedefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionFunction, definition, no evidence levelFunctiondefinitionPreorder, definition, no evidence levelPreorderdefinitionMonotone function, definition, no evidence levelMonotone functiondefinitionFubini's theorem on differentiation, statement, no evidence levelFubini's theorem on d…statement
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  1. Element definition
  2. Limit point of a set definition
  3. Proposition concept
  4. Set definition
  5. Limit of a function definition
  6. Implication definition
  7. Predicate definition
  8. Ordered pair definition
  9. Derivative definition
  10. Existential quantifier definition
  11. Universal quantifier definition
  12. Cartesian product definition
  13. Subset definition
  14. Uniqueness quantifier definition
  15. Binary relation definition
  16. Reflexive relation definition
  17. Transitive relation definition
  18. Function definition
  19. Preorder definition
  20. Monotone function definition
  21. Fubini's theorem on differentiation statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fubini%27s_theorem_on_differentiation
About this page
TagXDEJCKRVQ4
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Kindstatement, theorem
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AboutMonotone function
UsesMonotone function, Derivative
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Changed2 November 2026

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