VI.72 Henri Lebesgue
b. Beauvais, France, 1875; d. Paris, 1941
Theory of the integral; measure; applications in Fourier analysis;
dimension in topology; calculus of variations
Lebesgue studied at the École Normale in Paris (1894-97), where he was influenced by the slightly older BOREL [VI.70] and René-Louis Baire. As a teacher at Nancy he completed his seminal thesis “Intégrale, longueure, aire” (1902). After university positions in Rennes, Poitiers, and at the Sorbonne in Paris, and following war-related research, Lebesgue became a professor at the Sorbonne (1919) and then, finally, at the Collège de France (1921). One year later he was elected to the French Academy of Sciences.
Lebesgue’s most important achievement was his generalization of RIEMANN’S [VI.49] notion of an integral. This was partly in response to the need to include broader classes of real-valued functions, and partly to give secure foundations to concepts such as the interchangeability of limit and integral in infinite series (particularly Fourier series). Alluding to a famous example (1881) by Vito Volterra of a bounded derivative that could not be integrated, Lebesgue wrote in his thesis:
The kind of integration defined by Riemann does not allow in all cases for the solution of the fundamental problem of the calculus: find a function with a given derivative. It thus seems natural to search for a definition of the integral which makes integration the inverse operation of differentiation in as large a class of functions as possible.
Lebesgue defined his integral by partitioning the range of a function and summing up sets of x-coordinates (or arguments) belonging to given y-coordinates (or ordinates), rather than, as had traditionally been done, partitioning the domain. Lebesgue himself, according to his colleague, Paul Montel, compared his method with paying off a debt:
I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken out all my money I order the bills and coins according to identical values and then I pay the several heaps one after another to the creditor. This is my integral.
The comparison reveals the more theoretical character of Lebesgue’s integral, as compared with the more intuitive and natural summation used by Riemann. This meant that more sophisticated functions, which were not necessarily integrable in Riemann’s sense, became “summable” according to Lebesgue.
In order to perform his summations, Lebesgue had to base his new integral on Borel’s notion of MEASURE [III.55] (1898), which in turn drew heavily on CANTOR’S [VI.54] theory of infinite sets. He used infinitely many intervals to cover and to measure sets, and was thus able to measure much less intuitive subsets of the linear continuum (the reals) than had hitherto been considered. A crucial role was played by the notion of “the set of measure zero” and the consideration of properties that were valid “except for” such sets, i.e., “almost everywhere.” This allowed for the theory to be streamlined to include fundamental results such as: “A bounded function is Riemann integrable if and only if the set of its points of discontinuity has measure zero.”
Lebesgue completed Borel’s theory of measure, making it a true generalization of JORDAN’S [VI.52] earlier theory. From Jordan he also borrowed the important notion of a function of bounded variation for his theory of the integral. Lebesgue ascribed a measure to any subset of a “set of measure zero,” and opened up broader theoretical questions such as whether there exist any sets that are not Lebesgue-measurable. The latter question was proved in the affirmative by the Italian Giuseppe Vitali in 1905 with the help of the AXIOM OF CHOICE [III.1], while Robert Solovay showed in 1970, with methods of mathematical logic, that without the axiom of choice such existence cannot be proved (see SET THEORY [IV.22 §5.2]). Lebesgue himself remained skeptical about an unlimited use of set-theoretical principles such as the axiom of choice. He held a restrictive view of the “existence” of mathematical objects by making “definability” the touchstone for his empiricist philosophy of mathematics.
Lebesgue’s integral—the idea of which was paralleled, although not in such depth, in the work of the English mathematician W. H. Young—served as a sophisticated stimulus to developments in harmonic and functional analysis (e.g., the Lp spaces of RIESZ [VI.74] (1909)). Generalizations to functions defined on n-dimensional space, proposed by Lebesgue himself (1910), contributed to even more general theories of integrals, e.g., the theory of Radon (1913).
Although it took several decades for the importance of Lebesgue’s integral to become widely recognized, its significance for applications, especially in the analysis of discontinuous and statistical phenomena of nature and in probability theory, could not be ignored in the long run.
Further Reading
Hawkins, T. 1970. Lebesgue’s Theory of Integration: Its Origins and Development. Madison, WI: University of Wisconsin Press.
Lebesgue, H. 1972-73. Œuvres Scientifiques en Cinq Volumes. Geneva: Université de Genève.