Henri Lebesgue

How to use this course

Every graduate student meets Lebesgue’s name in the first week of real analysis: Lebesgue measure, the Lebesgue integral, the dominated convergence theorem, \(L^p\) spaces. Fewer know that the whole theory appeared in a six-page note and a thesis written while its author taught at a provincial lycée, or that its author spent his later years writing about elementary geometry and the teaching of measurement. This course presents the mathematics in the order Lebesgue built it, beside the story of a career that went from hostility to the Académie des Sciences in twenty years.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Where a theorem now carries Lebesgue’s name but reached its modern form through others, such as Vitali, Fatou or Riesz, the lesson says so.

Timeline

Beauvais to the École Normale, 1875–1897

Henri Léon Lebesgue was born in Beauvais, about 70 km north of Paris, on 28 June 1875. His father worked as a typesetter and died of tuberculosis when Henri was still a small child. His mother, a primary-school teacher, raised the family alone on little money.

At the Collège de Beauvais the boy stood out quickly. One of his teachers arranged support from the town so that he could continue in Paris, first at the Lycée Saint-Louis and then at the Lycée Louis-le-Grand, the classic route to the grandes écoles. In 1894 he entered the École Normale Supérieure.

At the ENS he passed the agrégation, the national teaching examination, in 1897. Instead of taking a school post at once, he spent the next two years working in the library of the École Normale. There he read the recent papers of René Baire on discontinuous functions and saw that much more could be done with them. He also studied the 1898 lectures on the theory of functions by Émile Borel, four years his senior, which contained the first countably additive measure on the line. Borel would supervise his thesis.

Nancy and the new integral, 1899–1902

In 1899 Lebesgue took a post teaching at the Lycée Central in Nancy. His research of the next three years was carried out alongside a full school timetable. His first paper (1898) concerned the Weierstrass approximation theorem. Between 1899 and 1901 he published six short notes in the Comptes Rendus of the Académie des Sciences.

Surfaces applicable to the plane

Most of the notes dealt with surfaces that can be unrolled onto a plane without stretching. Classical differential geometry said that smooth surfaces of this kind are ruled, like cones and cylinders. Lebesgue showed that once smoothness is dropped there are many more, such as surfaces obtained by crumpling a sheet of paper without stretching it. Problems like this pushed him to ask what area and length mean for surfaces and curves that are continuous but not differentiable, and that in turn required a better integral.

The note of 29 April 1901

The last note, Sur une généralisation de l’intégrale définie, defined the measure of a set and an integral built on it, and stated that the new integral agreed with Riemann’s when Riemann’s existed. It also observed that every bounded derivative is integrable in the new sense, something that fails for the Riemann integral. Lebesgue developed the theory in full in his thesis, Intégrale, longueur, aire, defended in Paris in 1902 and published the same year, 130 pages long, in the Milan journal Annali di Matematica Pura ed Applicata. It is one of the most consequential theses in the history of mathematics.

The reception in France was cool. Many classical analysts saw the subject as a study of monsters. The young theory was welcomed sooner by a few readers abroad, notably in Italy and later in Russia.

Rennes, Poitiers, Paris, 1902–1914

In 1902 Lebesgue became maître de conférences, a junior lecturer, at the Faculty of Science in Rennes. In the winter of 1902–03 he gave the Cours Peccot at the Collège de France, a lecture series for young mathematicians, and turned it into Leçons sur l’intégration et la recherche des fonctions primitives (1904), published in Borel’s series of short monographs. A second course in 1904–05 became Leçons sur les séries trigonométriques (1906). These two small books carried the theory to a wide audience.

The years also brought conflict. In 1904 he fell out with Baire over who had the stronger claim to give the Cours Peccot; the dispute later grew more bitter. In 1905 he joined Baire, Borel and Jacques Hadamard in an exchange of letters, published in the Bulletin de la Société Mathématique de France, on Ernst Zermelo’s proof of the well-ordering theorem. Lebesgue sided with Borel and Baire against arbitrary choices that cannot be described, a position that shaped his later work on definable sets (Lesson 10).

He moved to Poitiers in 1906 and became professor of mechanics there in 1907. In 1910 he returned to Paris as maître de conférences at the Sorbonne. The same year he published a long memoir extending integration and differentiation to several variables (Lesson 07). In 1911 he entered the debate on the invariance of dimension (Lesson 09). His work on Fourier series and on the Dirichlet problem continued. In 1914 the Académie des Sciences awarded him the Prix Poncelet, a sign that the French establishment had come round.

From content to measure

For the general reader. To integrate a function one must first know how big a set is. The length of an interval is obvious; the length of the set of rational numbers, or of a dust of points like the Cantor set, is not. Camille Jordan used finite covers, Borel allowed countable ones, and Lebesgue turned Borel’s idea into a complete theory.

Jordan content (1892)

For a bounded set \(E\subset\mathbb R\), Jordan’s outer content is the infimum of total lengths of finite unions of intervals covering \(E\), and the inner content is the supremum of total lengths of finite unions of intervals inside \(E\). \(E\) is Jordan measurable when the two agree. The set \(\mathbb Q\cap[0,1]\) has outer content 1 and inner content 0, so it has no content at all. A Riemann-integrable function is one whose region under the graph is Jordan measurable, which is why Riemann’s theory inherits the same weakness.

Lebesgue outer measure

Borel (1898) required countable additivity for sets built from intervals by countable unions and complements. Lebesgue defined, for any \(E\subset\mathbb R\),

For \(E\subseteq[a,b]\) he called \(E\) measurable when \(m^*(E)+m^*([a,b]\setminus E)=b-a\), that is, when outer and inner measure agree. Constantin Carathéodory’s 1914 criterion, now standard, says \(E\) is measurable when \(m^*(A)=m^*(A\cap E)+m^*(A\setminus E)\) for every \(A\subseteq\mathbb R\).

Proof sketch

The key estimate is that an interval \([a,b]\) has outer measure exactly \(b-a\). The inequality \(\le\) is trivial. For \(\ge\), given a countable cover by open intervals, compactness (the Heine–Borel theorem, which Borel proved for exactly this purpose) extracts a finite subcover, and a finite cover of \([a,b]\) has total length at least \(b-a\). Countable subadditivity of \(m^*\) follows from an \(\varepsilon/2^k\) argument. With Carathéodory’s criterion, closure under complements is immediate, and closure under countable unions together with additivity follows by splitting an arbitrary test set \(A\) along the \(E_k\).

A set that cannot be measured

Call \(x\sim y\) when \(x-y\in\mathbb Q\), and choose one representative in \([0,1]\) of each class to form \(V\). The translates \(V+q\), \(q\in\mathbb Q\cap[-1,1]\), are disjoint, cover \([0,1]\), and lie inside \([-1,2]\). If \(\mu(V)=0\) their union has measure 0; if \(\mu(V)>0\) it has infinite measure. Both contradict \(1\le\mu\le3\). Lebesgue distrusted such non-constructive sets, but the result confirms that his choice of a restricted class \(\mathcal L\) was unavoidable.

The Lebesgue integral

For the general reader. Riemann cut the horizontal axis into small pieces and added up thin vertical rectangles. Lebesgue cut the vertical axis instead. For each band of values he asked how large the set of points is where the function takes values in that band, and multiplied. Lebesgue liked to compare this to counting money by first sorting the coins by denomination, rather than adding them in the order they come out of the purse.

Lebesgue’s definition

A function \(f:[a,b]\to\mathbb R\) is measurable if \(\{x: f(x)>c\}\in\mathcal L\) for every \(c\). Let \(f\) be bounded and measurable with \(m\le f\lt M\). For a partition \(m=y_0\lt y_1\lt \dots\lt y_n=M\) of the range, put \(E_k=\{x: y_{k-1}\le f(x)\lt y_k\}\) and form

The \(E_k\) are disjoint and cover \([a,b]\), so as the mesh of the partition goes to 0, \(s\) and \(S\) converge to a common limit, the Lebesgue integral \(\int_a^b f\). No continuity is needed, only measurability.

The modern form

A simple function is \(\varphi=\sum_{k} c_k \mathbf 1_{A_k}\) with finitely many measurable \(A_k\), and \(\int\varphi=\sum_k c_k\, m(A_k)\). For measurable \(f\ge0\) one sets \(\int f=\sup\{\int\varphi : 0\le\varphi\le f,\ \varphi \text{ simple}\}\), and \(f\) is integrable when \(\int|f|\lt \infty\), with \(\int f=\int f^+-\int f^-\). For \(f\ge0\) this gives the layer-cake formula, which is Lebesgue’s picture in one line:

Proof sketch of (2)

Let \(\omega(x)=\lim_{\delta\to0}\big(\sup_{B_\delta(x)} f-\inf_{B_\delta(x)} f\big)\) be the oscillation, so the discontinuity set is \(D=\bigcup_n D_n\) with \(D_n=\{\omega\ge1/n\}\), each \(D_n\) closed. If \(m(D)=0\), each compact \(D_n\) is covered by finitely many intervals of small total length; on the rest of \([a,b]\) the oscillation is below \(1/n\), and the upper and lower Riemann sums differ by at most \((b-a)/n+2\sup|f|\cdot\varepsilon\). Conversely, if \(m(D_n)>0\) for some \(n\), every partition has intervals meeting \(D_n\) in their interiors with total length at least \(m(D_n)\), each contributing oscillation at least \(1/n\), so upper minus lower sums stay above \(m(D_n)/n\).

Limits under the integral sign

For the general reader. Analysts constantly need to swap a limit and an integral. Under Riemann’s theory this required uniform convergence, a strong and often unavailable condition. Lebesgue’s integral allows the swap under a simple domination hypothesis, and the payoff is that spaces of integrable functions become complete, the property that makes modern functional analysis possible.

Let \((f_n)\) be measurable functions on a measure space, for instance an interval with Lebesgue measure.

Lebesgue proved the bounded case on an interval (\(|f_n|\le M\)) in his thesis; this already contained the key idea, and the version with an integrable dominating function followed in his later work. Each statement implies the next with little effort: Fatou follows from monotone convergence applied to \(\inf_{k\ge n} f_k\), and dominated convergence follows from Fatou applied to \(2g-|f_n-f|\ge0\), which gives \(\int 2g\le\int 2g-\limsup\int|f_n-f|\).

Completeness

For \(1\le p\lt \infty\), \(L^p\) is the space of measurable \(f\) with \(\|f\|_p=\big(\int|f|^p\big)^{1/p}\lt \infty\), functions equal almost everywhere being identified.

Frigyes Riesz and Ernst Fischer proved it independently in 1907. The proof picks a subsequence with \(\|f_{n_{k+1}}-f_{n_k}\|\lt 2^{-k}\), uses monotone convergence to show \(\sum_k|f_{n_{k+1}}-f_{n_k}|\) is finite almost everywhere, and defines the limit pointwise. With the Riemann integral, \(L^2\) is not complete, and the isomorphism of \(L^2\) with \(\ell^2\) that quantum mechanics would use twenty years later does not exist.

Differentiation and primitives

For the general reader. The fundamental theorem of calculus says that integration and differentiation undo each other. Lebesgue asked how far this survives for rough functions. His answer was that every monotone function has a derivative almost everywhere, and that an integral can always be differentiated back, almost everywhere. The subtitle of his 1904 book, the search for primitive functions, was this problem.

Lebesgue proved (1) for continuous monotone functions; the general case followed soon after. The exact class of functions that are integrals of their derivatives is the absolutely continuous functions, a term introduced by Giuseppe Vitali in 1905: \(F\) is absolutely continuous if for every \(\varepsilon>0\) there is \(\delta>0\) such that \(\sum|F(b_i)-F(a_i)|\lt \varepsilon\) for every finite family of disjoint intervals with \(\sum(b_i-a_i)\lt \delta\).

Several variables (1910)

In his 1910 memoir on the integration of discontinuous functions, Lebesgue extended differentiation to \(\mathbb R^n\), replacing intervals by regular families of sets shrinking to a point. The modern statement uses balls.

Proof idea. The statement is clear for continuous \(f\), which are dense in \(L^1\). To pass to the limit one needs control of the maximal function \(Mf(x)=\sup_r\frac{1}{|B_r|}\int_{B_r(x)}|f|\). A covering lemma of Vitali type gives the weak-type bound \(|\{Mf>\lambda\}|\le\frac{3^n}{\lambda}\|f\|_1\), and an approximation argument finishes the proof. This packaging is due to G. H. Hardy, J. E. Littlewood and Norbert Wiener in the 1930s; Lebesgue’s own argument used Vitali’s covering theorem directly.

Density theorem. Applying the theorem to \(f=\mathbf 1_E\): almost every point of a measurable set \(E\) is a point of density 1, meaning \(|E\cap B_r(x)|/|B_r(x)|\to1\), and almost every point outside has density 0. Measurable sets are, at small scales, almost everywhere either full or empty.

Trigonometric series

For the general reader. Joseph Fourier claimed that every function is a sum of sines and cosines. Much of nineteenth-century analysis grew from trying to make this precise, and Riemann invented his integral for this purpose. Lebesgue’s integral gave the subject its natural setting. He proved that Fourier coefficients always decay, explained exactly why partial sums can misbehave, and showed that averaged partial sums recover almost every function almost everywhere.

For \(f\in L^1(-\pi,\pi)\) let \(\hat f(k)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-ikt}\,dt\) and \(S_nf(x)=\sum_{|k|\le n}\hat f(k)e^{ikx}\). Then

Proof. For the indicator of an interval \([a,b]\), \(|\hat f(k)|=\frac{|e^{-ikb}-e^{-ika}|}{2\pi|k|}\le\frac{1}{\pi|k|}\). By linearity the result holds for step functions, which are dense in \(L^1\). Since \(|\hat f(k)-\hat g(k)|\le\frac{1}{2\pi}\|f-g\|_1\), it passes to all of \(L^1\).

The Lebesgue constants

The norm of \(S_n\) as an operator on continuous functions with the supremum norm is the Lebesgue constant

Because \(L_n\to\infty\), the uniform boundedness principle of Stefan Banach and Hugo Steinhaus (1927) gives a continuous function whose Fourier series diverges at a point, an example Paul du Bois-Reymond had constructed by hand in 1873. Lebesgue’s analysis of \(L_n\), and his 1905 study of the classical Dini–Lipschitz and Jordan tests, showed precisely what the partial sums can and cannot do. The same constants measure the stability of polynomial interpolation, where the name “Lebesgue constant” is now most common.

Fejér means

Lipót Fejér proved in 1900 that the averages \(\sigma_nf=\frac{1}{n+1}\sum_{j=0}^{n}S_jf\) converge to \(f\) at every point of continuity. They are given by a positive kernel, \(\sigma_nf=\frac{1}{2\pi}f*K_n\) with \(K_n(t)=\frac{1}{n+1}\Big(\frac{\sin((n+1)t/2)}{\sin(t/2)}\Big)^2\ge0\).

Whether \(S_nf\) itself converges almost everywhere for \(f\in L^2\) remained open until Lennart Carleson proved it in 1966. Andrey Kolmogorov had shown in 1923 that for \(L^1\) it can fail.

Dimension, coverings, the Dirichlet problem

For the general reader. How do we know that a plane and a solid are different, so that no continuous one-to-one map turns one into the other? Lebesgue saw a simple answer in the way bricks are laid: if you tile a region of the plane with small bricks, some point must touch three of them, while in space some point must touch four. The same years saw him probe the limits of potential theory with a famous thorn-shaped domain.

Invariance of dimension

In 1911 Lebesgue published a note in Mathematische Annalen sketching why \(\mathbb R^m\) and \(\mathbb R^n\) are not homeomorphic for \(m\ne n\). L. E. J. Brouwer, who had just proved the theorem by different methods, objected that the sketch had gaps. Brouwer supplied a full proof of Lebesgue’s covering principle in 1913, and the episode left lasting bad feeling. The principle itself became the basis of dimension theory.

This motivates the covering dimension: a compact metric space \(X\) has \(\dim X\le n\) if every open cover has a finite open refinement in which every point lies in at most \(n+1\) sets. The theorem says \(\dim[0,1]^n=n\), and since covering dimension is a topological invariant, cubes of different dimensions are not homeomorphic.

The Lebesgue spine

The Dirichlet problem asks for a harmonic function in a domain \(\Omega\) with prescribed continuous boundary values. Stanisław Zaremba showed in 1911 that it can fail for a punctured disc. Soon after, Lebesgue gave a sharper example in three dimensions: remove from a ball a very thin closed cusp, a spine, ending at a point \(p\). If the spine is sharp enough, for example of the form \(r\le e^{-1/z}\) near the tip at the origin, the generalized solution does not take the prescribed value at \(p\). The boundary point \(p\) is irregular, even though \(\Omega\) is connected and its complement near \(p\) is a solid cusp, not a single point. Lebesgue built it from the potential of a line charge whose density grows linearly along a segment, whose level surfaces near the tip have exactly this shape. Norbert Wiener’s criterion (1924) later characterized regular points by the capacity of the complement near \(p\).

Analytic representation and Borel sets

For the general reader. Which functions can be written down by a formula? Lebesgue took this old question seriously. In 1905 he showed that the functions obtainable from continuous ones by repeated limits form an infinite hierarchy, every level of which is non-empty. In the same paper he made a mistake about projections of sets that, twelve years later, gave birth to descriptive set theory.

Baire classes

Baire defined class 0 as the continuous functions \(\mathbb R\to\mathbb R\), and class \(\alpha\) as pointwise limits of sequences of functions from earlier classes, not already in an earlier class, for every countable ordinal \(\alpha\). Lebesgue called the union of all classes the functions representable analytically.

Part (2) uses a diagonal argument with a function universal for each class, an early example of the hierarchy theorems that recur throughout logic and computability. Felix Hausdorff later gave the full correspondence between the Baire classes of functions and the levels of the Borel hierarchy of sets.

A fruitful error

In the same memoir Lebesgue asserted, with a short and incorrect argument, that the projection onto the \(x\)-axis of a Borel subset of the plane is again Borel. In 1917 Mikhail Suslin, a student of Nikolai Luzin in Moscow, found the error. Projections of Borel sets form a strictly larger class, the analytic sets, all of which are Lebesgue measurable. Suslin proved that a set is Borel exactly when it and its complement are both analytic. Luzin’s school built descriptive set theory on this. Lebesgue wrote the preface to Luzin’s Leçons sur les ensembles analytiques (1930), acknowledging that the subject had grown out of his own error.

The Collège de France, 1914–1941

During the First World War Lebesgue worked on problems for national defence. Borel was engaged in similar work, and the two men, once close, fell out during this period. Between 1918 and 1920 the quarrel became public: in the Annales de l’École Normale Supérieure they disputed whose integral was the more general and who deserved credit for measure theory. The friendship did not recover.

At the Sorbonne he became professor of the application of geometry to analysis at the end of the war. In 1921 he left for the Collège de France as Professor of Mathematics, the chair he held for the rest of his life. On 29 May 1922 he was elected to the Académie des Sciences. The same year he published a 92-page Notice on his own work, describing nearly ninety books and papers. He also taught for years at the École Supérieure de Physique et de Chimie Industrielles in Paris and at the École Normale Supérieure for women at Sèvres.

Students and style

The Mathematics Genealogy Project lists seven doctoral students, among them Paul Montel (1907), the Polish topologist Zygmunt Janiszewski (1911), Louis Antoine (1921) and Georges de Rham (1931). Antoine had been blinded in the war; Lebesgue encouraged him to work in topology, where he discovered the wild embedding now called Antoine’s necklace. De Rham’s thesis on the topology of manifolds led to de Rham cohomology.

After 1922 Lebesgue turned increasingly to pedagogy, history and elementary geometry. A long series of articles, Sur la mesure des grandeurs, in L’Enseignement Mathématique (1931–1935), examined how area, volume and length should be taught to schoolchildren and future teachers. Books on conics and on geometric constructions appeared after his death. He disliked generality for its own sake and once warned that mathematics reduced to general theories would be a beautiful form without content.

Last years and legacy

Lebesgue continued to lecture at the Collège de France through the 1930s. He died in Paris on 26 July 1941, during the German occupation, at the age of 66.

Legacy

• Analysis: the Lebesgue integral is the standard integral of mathematics. Its convergence theorems and the completeness of \(L^p\) underlie Banach and Hilbert space theory, distributions, Sobolev spaces and the modern theory of partial differential equations.
• Probability: Andrey Kolmogorov’s axioms of 1933 identify probability with a measure of total mass 1 and expectation with the Lebesgue integral.
• Physics: the state space of quantum mechanics is \(L^2\), complete only because of Lebesgue’s integral.
• Harmonic analysis and topology: the Riemann–Lebesgue lemma, Lebesgue constants, Lebesgue points, covering dimension and the Lebesgue number lemma remain in daily use.
• Honours: Prix Poncelet (1914) and other prizes of the Académie des Sciences; member of the Académie (1922); honorary member of the London Mathematical Society (1924); Foreign Member of the Royal Society (1934); membership of the academies of Belgium, Denmark, Italy (Lincei), Romania and Kraków, among others. A lunar crater bears his name.

Connections

People linked to Henri Lebesgue across this collection. A name without a link has no page here yet. See Henri Lebesgue in the network of all 337 lives →

Teachers and advisors

• Émile Borel 1902

Students

• André Marchaud 1927
• Florin Vasilesco 1925
• Georges de Rham 1931
• Louis Antoine 1921
• Paul Montel 1907
• Zygmunt Janiszewski 1911

Influenced by

• Bernhard Riemann
• Camille Jordan
• Thomas Joannes Stieltjes
• René Baire

Influenced

• Andrey Kolmogorov
• Frigyes Riesz
• Maurice Fréchet
• Norbert Wiener
• Stefan Banach
• Nikolai Luzin
• Pierre Fatou

Colleagues and collaborators

• Jacques Hadamard
• Wacław Sierpiński
• Giuseppe Vitali

Rivals

• L. E. J. Brouwer 1911-1913

Also mentioned in this life

• Georg Cantor
• Peter Gustav Lejeune Dirichlet
• Joseph Fourier
• G. H. Hardy
• Felix Hausdorff
• David Hilbert
• Jean-Victor Poncelet
• Karl Weierstrass

Fields

• real analysis 10 lives
• measure theory 5 lives
• topology 9 lives
• potential theory 9 lives

On CoursesHub

• Signal Theory
• Topology in Physics · Differential Geometry

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Henri Léon Lebesgue, MacTutor History of Mathematics.
2. Henri Lebesgue, Wikipedia.
3. Henri Léon Lebesgue, Mathematics Genealogy Project.
4. T. Hawkins, Lebesgue, Henri Léon, in Dictionary of Scientific Biography, Scribner, 1970–1990.
5. H. Lebesgue, Notice sur les travaux scientifiques de M. Henri Lebesgue, Toulouse, 1922.

Mathematics

1. H. Lebesgue, Sur une généralisation de l’intégrale définie, C. R. Acad. Sci. Paris 132 (1901).
2. H. Lebesgue, Intégrale, longueur, aire, Annali di Matematica Pura ed Applicata (3) 7 (1902).
3. H. Lebesgue, Leçons sur l’intégration et la recherche des fonctions primitives, Gauthier-Villars, 1904; 2nd ed. 1928.
4. H. Lebesgue, Leçons sur les séries trigonométriques, Gauthier-Villars, 1906.
5. H. Lebesgue, Sur les fonctions représentables analytiquement, J. Math. Pures Appl. (6) 1 (1905).
6. H. Lebesgue, Sur l’intégration des fonctions discontinues, Ann. Sci. ÉNS 27 (1910).
7. J. Hadamard, R. Baire, H. Lebesgue, E. Borel, Cinq lettres sur la théorie des ensembles, Bull. SMF 33 (1905).
8. T. Hawkins, Lebesgue’s Theory of Integration: Its Origins and Development, University of Wisconsin Press, 1970; AMS Chelsea reprint.
9. E. M. Stein and R. Shakarchi, Real Analysis: Measure Theory, Integration, and Hilbert Spaces, Princeton, 2005.
10. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, 1987.
11. A. S. Kechris, Classical Descriptive Set Theory, Springer, 1995.