Inmathematics, theLebesgue differentiation theorem is a theorem ofreal analysis, which states that for almost every point, the value of an integrable function is the limiting average taken around the point. The theorem is named forHenri Lebesgue.

For aLebesgue integrable real or complex-valued functionf onRⁿ, the indefinite integral is aset function which maps a measurable setA to the Lebesgue integral of${\displaystyle f\cdot {\mathbf{1}}_A}$, where${\displaystyle {\mathbf{1}}_A}$ denotes thecharacteristic function of the setA. It is usually written$${\displaystyle A\mapsto {\int }_{A}f\mathrm{d}\lambda ,}$$ withλ then–dimensionalLebesgue measure.

Thederivative of this integral atx is defined to be$${\displaystyle \underset{B\to x}{lim}\frac{1}{|B|}{\int }_{B}f\mathrm{d}\lambda ,}$$where |B| denotes the volume (i.e., the Lebesgue measure) of aballB  centered atx, andB →x means that the diameter ofB  tends to 0.
TheLebesgue differentiation theorem (Lebesgue 1910) states that this derivative exists and is equal tof(x) atalmost every pointx ∈Rⁿ.^[1] In fact a slightly stronger statement is true. Note that:$${\displaystyle \left|\frac{1}{|B|}{\int }_{B}f(y)\mathrm{d}\lambda (y)-f(x)\right|=\left|\frac{1}{|B|}{\int }_B(f(y)-f(x))\mathrm{d}\lambda (y)\right|\le \frac{1}{|B|}{\int }_B|f(y)-f(x)|\mathrm{d}\lambda (y).}$$

The stronger assertion is that the right hand side tends to zero for almost every pointx. The pointsx for which this is true are called theLebesgue points off.

A more general version also holds. One may replace the ballsB  by a family${\displaystyle \mathcal{V}}$ of setsU  ofbounded eccentricity. This means that there exists some fixedc > 0 such that each setU  from the family is contained in a ballB  with${\displaystyle |U|\ge c|B|}$. It is also assumed that every pointx ∈Rⁿ is contained in arbitrarily small sets from${\displaystyle \mathcal{V}}$. When these sets shrink tox, the same result holds: for almost every pointx,$${\displaystyle f(x)=\underset{U\to x,U\in \mathcal{V}}{lim}\frac{1}{|U|}{\int }_{U}f\mathrm{d}\lambda .}$$

The family of cubes is an example of such a family${\displaystyle \mathcal{V}}$, as is the family${\displaystyle \mathcal{V}}$(m) of rectangles inR² such that the ratio of sides stays betweenm⁻¹ andm, for some fixedm ≥ 1. If an arbitrary norm is given onRⁿ, the family of balls for the metric associated to the norm is another example.

The one-dimensional case was proved earlier byLebesgue (1904). Iff is integrable on the real line, the function$${\displaystyle F(x)={\int }_{(-\mathrm{\infty },x]}f(t)\mathrm{d}t}$$is almost everywhere differentiable, with${\displaystyle F^{\prime }(x)=f(x).}$ Were${\displaystyle F}$ defined by aRiemann integral this would be essentially thefundamental theorem of calculus, but Lebesgue proved that it remains true when using the Lebesgue integral.^[2]

The theorem in its stronger form—that almost every point is a Lebesgue point of alocally integrable functionf—can be proved as a consequence of theweak–L¹ estimates for theHardy–Littlewood maximal function. The proof below follows the standard treatment that can be found inBenedetto & Czaja (2009),Stein & Shakarchi (2005),Wheeden & Zygmund (1977) andRudin (1987).

Since the statement is local in character,f can be assumed to be zero outside some ball of finite radius and hence integrable. It is then sufficient to prove that the set

${\displaystyle E_{\alpha }=\{x\in {\mathbf{R}}^n:\underset{|B|\to 0,x\in B}{lim sup}\frac{1}{|B|}|{\int }_{B}f(y)-f(x)\mathrm{d}y|>2\alpha \}}$

has measure 0 for allα > 0.

Letε > 0 be given. Using thedensity ofcontinuous functions ofcompactsupport inL¹(Rⁿ), one can find such a functiong satisfying

It is then helpful to rewrite the main difference as

${\displaystyle \frac{1}{|B|}{\int }_{B}f(y)\mathrm{d}y-f(x)=(\frac{1}{|B|}{\int }_B(f(y)-g(y))\mathrm{d}y)+(\frac{1}{|B|}{\int }_{B}g(y)\mathrm{d}y-g(x))+(g(x)-f(x)).}$

The first term can be bounded by the value atx of the maximal function forf − g, denoted here by${\displaystyle (f-g{)}^{\ast }(x)}$:

${\displaystyle \frac{1}{|B|}{\int }_B|f(y)-g(y)|\mathrm{d}y\le \underset{r>0}{sup}\frac{1}{|B_r(x)|}{\int }_{B_r(x)}|f(y)-g(y)|\mathrm{d}y=(f-g{)}^{\ast }(x).}$

The second term disappears in the limit sinceg is a continuous function, and the third term is bounded by |f(x) −g(x)|. For the absolute value of the original difference to be greater than 2α in the limit, at least one of the first or third terms must be greater thanα in absolute value. However, the estimate on the Hardy–Littlewood function says that

for some constantA_n depending only upon the dimensionn. TheMarkov inequality (also called Tchebyshev's inequality) says that

thus

${\displaystyle |E_{\alpha }|\le \frac{A_n+1}{\alpha }\epsilon .}$

Sinceε was arbitrary, it can be taken to be arbitrarily small, and the theorem follows.

TheVitali covering lemma is vital to the proof of this theorem; its role lies in proving the estimate for theHardy–Littlewood maximal function.

The theorem also holds if balls are replaced, in the definition of the derivative, by families of sets with diameter tending to zero satisfying theLebesgue's regularity condition, defined above asfamily of sets with bounded eccentricity. This follows since the same substitution can be made in the statement of the Vitali covering lemma.

This is an analogue, and a generalization, of thefundamental theorem of calculus, which equates aRiemann integrable function and the derivative of its (indefinite) integral. It is also possible to show a converse – that every differentiable function is equal to the integral of its derivative, but this requires aHenstock–Kurzweil integral in order to be able to integrate an arbitrary derivative.

A special case of the Lebesgue differentiation theorem is theLebesgue density theorem, which is equivalent to the differentiation theorem for characteristic functions of measurable sets. The density theorem is usually proved using a simpler method (e.g. see Measure and Category).

This theorem is also true for every finite Borel measure onRⁿ instead of Lebesgue measure (a proof can be found in e.g. (Ledrappier & Young 1985)). More generally, it is true of any finite Borel measure on a separable metric space such that at least one of the following holds:

• the metric space is aRiemannian manifold,
• the metric space is alocally compactultrametric space,
• the measure isdoubling.

A proof of these results can be found in sections 2.8–2.9 of (Federer 1969).

• Lebesgue's density theorem

• Lebesgue, Henri (1904).Leçons sur l'Intégration et la recherche des fonctions primitives. Paris: Gauthier-Villars.
• Lebesgue, Henri (1910)."Sur l'intégration des fonctions discontinues".Annales Scientifiques de l'École Normale Supérieure.27:361–450.doi:10.24033/asens.624.
• Wheeden, Richard L.;Zygmund, Antoni (1977).Measure and Integral – An introduction to Real Analysis. Marcel Dekker.
• Oxtoby, John C. (1980).Measure and Category. Springer Verlag.
• Stein, Elias M.; Shakarchi, Rami (2005).Real analysis. Princeton Lectures in Analysis, III. Princeton, NJ: Princeton University Press. pp. xx+402.ISBN 0-691-11386-6.MR 2129625
• Benedetto, John J.; Czaja, Wojciech (2009).Integration And Modern Analysis. Birkhäuser Advanced Texts. Springer. pp. 361–364.ISBN 978-0817643065.
• Rudin, Walter (1987).Real and complex analysis. International Series in Pure and Applied Mathematics (3rd ed.). McGraw–Hill.ISBN 0070542341.
• Ledrappier, F.;Young, L.S. (1985). "The Metric Entropy of Diffeomorphisms: Part I: Characterization of Measures Satisfying Pesin's Entropy Formula".Annals of Mathematics.122 (3):509–539.doi:10.2307/1971328.JSTOR 1971328.
• Federer, Herbert (1969).Geometric measure theory. Die Grundlehren der mathematischen Wissenschaften, Band. Vol. 153. New York: Springer-Verlag New York Inc.

• Theorems in real analysis
• Theorems in measure theory

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