Math 2.0

Measure and integration Differentiation theory Theorem

Lebesgue differentiation theorem

Builds on Lebesgue measureIntegrable functionAlmost everywhereOpen ballLebesgue pointHardy–Littlewood maximal functionVitali covering lemma

Statement

Let n≥1n \ge 1, let λ\lambda be Lebesgue measure on ℝn\R^n, and let f ⁣:ℝn→ℝf \colon \R^n \to \R be locally integrable, that is, integrable on every bounded measurable set. For x∈ℝnx \in \R^n and r>0r > 0, let B(x,r)B(x, r) be the open ball of centre xx and radius rr. Then for almost every x∈ℝnx \in \R^n, lim⁡r→0+1λ(B(x,r))∫B(x,r)∣f(y)−f(x)∣ dλ(y)=0.\lim_{r \to 0^+} \frac{1}{\lambda(B(x, r))} \int_{B(x, r)} \abs{f(y) - f(x)} \, d\lambda(y) = 0. In particular, lim⁡r→0+1λ(B(x,r))∫B(x,r)f dλ=f(x)\lim_{r \to 0^+} \frac{1}{\lambda(B(x, r))} \int_{B(x, r)} f \, d\lambda = f(x) for almost every xx.

Remarks

The conclusion says that almost every point is a Lebesgue point of ff. The proof first notes that the claim is clear for continuous ff with compact support, which are dense in L1L^1. The error for general ff is controlled by the Hardy–Littlewood maximal function, which satisfies the weak type bound λ({Mf>t})≤3n∥f∥1/t\lambda(\set{Mf > t}) \le 3^n \norm{f}_1 / t. This bound comes from the Vitali covering lemma.

Example: for n=1n = 1 and ff integrable on ℝ\R, the theorem says F(x)=∫0xfF(x) = \int_0^x f satisfies F′(x)=f(x)F'(x) = f(x) for almost every xx.

The balls can be replaced by other families that shrink nicely, such as cubes containing xx. They cannot be replaced by arbitrary rectangles containing xx: in dimension n≥2n \ge 2, for rectangles of unbounded eccentricity the conclusion can fail for some integrable ff.

Going deeper

For a student

The theorem is the measure-theoretic form of the fundamental theorem of calculus. It gives Lebesgue's density theorem when ff is an indicator function. It extends to Radon measures on ℝn\R^n through the Besicovitch covering theorem, and to doubling metric measure spaces.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Almost everywhere, concept, no evidence levelAlmost everywhereconceptBorel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptElement, definition, no evidence levelElementdefinitionEuclidean space, object, no evidence levelEuclidean spaceobjectHardy–Littlewood maximal function, concept, no evidence levelHardy–Littlewood maxi…conceptIntegrable function, concept, no evidence levelIntegrable functionconceptLebesgue point, concept, no evidence levelLebesgue pointconceptMetric space, concept, no evidence levelMetric spaceconceptOpen ball, concept, no evidence levelOpen ballconceptOuter measure, concept, no evidence levelOuter measureconceptProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSeparable space, concept, no evidence levelSeparable spaceconceptSet, definition, no evidence levelSetdefinitionLebesgue measure, object, no evidence levelLebesgue measureobjectImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionExistential 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 relationdefinitionFunction, definition, no evidence levelFunctiondefinitionMetric space, definition, no evidence levelMetric spacedefinitionClosed ball, definition, no evidence levelClosed balldefinitionVitali covering lemma, statement, no evidence levelVitali covering lemmastatementLebesgue differentiation theorem, statement, no evidence levelLebesgue differentiat…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Almost everywhere concept
  2. Borel sigma-algebra concept
  3. Element definition
  4. Euclidean space object
  5. Hardy–Littlewood maximal function concept
  6. Integrable function concept
  7. Lebesgue point concept
  8. Metric space concept
  9. Open ball concept
  10. Outer measure concept
  11. Proposition concept
  12. Real numbers object
  13. Separable space concept
  14. Set definition
  15. Lebesgue measure object
  16. Implication definition
  17. Predicate definition
  18. Ordered pair definition
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Subset definition
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Function definition
  26. Metric space definition
  27. Closed ball definition
  28. Vitali covering lemma statement
  29. Lebesgue differentiation theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Lebesgue_differentiation_theorem
About this page
TagSAZ0ESVBVP
URIm2://SAZ0ESVBVP
Kindstatement, theorem
WritingW0 stub
MSC28A15, 42B25
StatusThis copy of the site has no build data. Run m2 status SAZ0ESVBVP.
AboutLebesgue point
UsesLebesgue measure, Integrable function, Almost everywhere, Open ball, Lebesgue point, Hardy–Littlewood maximal function, Vitali covering lemma
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Changed2 November 2026

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