Lebesgue's New Integration Theory

Date 1900

Henri-Léon Lebesgue developed a new theory for integrating discontinuous functions, based on a more general set-theoretic concept of measure, which furthered the development of the calculus.

Locale Rennes, France

Key Figures

• Henri-Léon Lebesgue (1875-1941), French mathematician
• Émile Borel (1871-1956), French mathematician and politician
• William Henry Young (1863-1943), English mathematician

Summary of Event

Since Euclid’s Stoicheia (c. 300 b.c.e.; The Elements of Geometrie of the Most Auncient Philosopher Euclide of Megara, 1570; commonly known as the Elements), the theory of measurements in mathematics generally was thought to encompass little more than systematically comparing the points, lines, or planes to be measured to a standard reference. With Pythagoras’s discovery of geometric incommensurables (irrational numbers), it was realized gradually that the question of mathematical measurement, in general, requires more precise and comprehensive consideration of seemingly infinite processes and collections. Development of the differential and integral calculus and limit theory by Sir Isaac Newton and Gottfried Wilhelm Leibniz brought with it the realization that, for most geometric figures, true mathematical measures do not exist a priori, but rather depend on the existence and computability of strictly defined associated limits.

In 1822, French physicist and mathematicianJoseph Fourier discovered that computation of sets of harmonic (trigonometric) series used to approximate a given function depended upon appropriate existence and calculations using integrals. Integration, or integral theory, concerns the techniques of finding a function g(x), the first derivatives of which are equal to a given function f(x). These, in turn, depend on how discontinuous the function is. A function is a mathematical expression defining the relation between one (independent) and another (dependent) variable. Although it was known that every continuous function has an integral summation, it was not clear at that time whether or how an integral could be defined for the many different classes of discontinuous functions; that is, those functions that are not definable at one or more specific points. A function is continuous if it is possible to plot it as a single, unbroken curve.

In 1854, the German mathematician Georg Friedrich Riemann offered the first partial answer to the question of how to integrate discontinuous functions, based on approximating integrands having only a finite number of definitely known discontinuous points by a sum of step functions, instead of the curve-tangent sums of earlier calculus. The sum or measure of the Riemann integral is equal to the area of the region bounded by the curve f(x). Yet, one of the many classic functions of importance to mathematics and physics, for which Riemann integration cannot be defined, is the “salt and pepper” function of Peter Dirichelet, where f(x) = 1 if x is rational, and 0 if x is irrational.

Earlier, in 1834, Austrian mathematician-philosopher Bernhard Bolzano gave examples of mathematically continuous functions that are nowhere differentiable and, thus, unintegrable by Riemann’s definition. Karl Theodor Weierstrass provided similar examples in 1875. Further motivations for clarifying the notions of continuity and integration arose in 1885, when German mathematician Adolf Harnack paradoxically showed that any countable subset of the real number system could be covered by a collection of intervals of arbitrarily small total length.

As a reaction to these difficulties, between 1880 and 1885, French geometrician Jean Darboux gave a novel definition of continuity, for the first time as a locally definable (versus global) mathematical property for discontinuous functions. Likewise, Camille Jordan, in 1892, first defined analogously the more general notion of “mathematical measure,” using finite unions of mathematical intervals to approximate sparse and dense subsets of real numbers. Nevertheless, the opinions of many leading mathematicians such as Henri Poincaré and Charles Hermite differed as to whether discontinuous functions and functions without derivatives were legitimate mathematical objects, as well as how to define the concept of a normal versus a “pathological” function.

The first mathematician to infer from the above results that countable unions of intervals should be used to measure the more general entity of real number subsets was Émile Borel. In 1898, in his Leçons sur la théorie des fonctions (lectures on the theory of functions), Borel advocated an abstract axiomatics of constructivistic definitions. Constructivistic, in this case, meant that all proposed definitions should permit explicit construction of actual examples of the mathematical entities referred to. Borel redefined the “measure” of any countable union of real number intervals to be its total length and thereby extended the notion of abstract measurability to progressively more complex sets. Borel sought to generalize Georg Cantor’s set theory, as well as to explicitly study “pathological” functions definable in terms of point sets. For Borel, the main problem was how to assign consistently to each pathologic point or singularity an appropriate numerical measure, meaning a nonnegative real number precisely analogous to length, area, and volume. Starting with elementary geometrical figures, Borel sought to define constructively measures to these sets so that formal measures of a line segment, or polygon, is always the same as its Euclidean measure and that the measure of a finite or countably infinite union of non-overlapping sets is equal to the sum of the measures of all individual sets.

Cantor’s set theory had expanded the definition of continuity to include not only geometric smoothness (or nonvariability) of a curve but also its pointwise mapping, or set theoretic correspondence. One of the results of Borel’s studies was the well-known Heine-Borel theorem, which states that if a closed set of points on a line can be covered by a set of intervals, such that every point of this set is an interior point of at least one of the intervals, then there exists a finite number of intervals with this“covering” property. For Borel, any such set obtainable by the basic mathematical properties and operations of union and intersection of sets in principle has a measure.

With these ideas as background, in 1900, Henri-Léon Lebesgue sought to enlarge Borel’s notion of measurable sets in order to apply it explicitly to the problem of integrating a wider class of pathological functions than those permitted by Riemann’s integral. In the preface of his doctoral dissertation, Lebesgue outlined his motivations and methods. In contrast to Borel, Lebesgue employed a nonaxiomatic descriptive approach, one of the key results of which was to solve the problem of defining an integral measure for discontinuous functions in general, insofar as it is necessary here that an infinite but bounded set have finite measure. Lebesgue generalized Riemann’s definition of the integral by applying this new definition of measure.

In the second chapter of his dissertation, Lebesgue proposed five criteria necessary for sufficiently widening integration theory, including the need to contain Riemann’s definition as a special case to incorporate only assumptions and results in this extension that are natural, necessary, and computationally useful. Another key insight of Lebesgue’s integration theory is that every function with bounded measure is also integrable. Perhaps the most critical property of Borel-Lebesgue measure is its property of summability, or countable additivity. In particular, Lebesgue showed that this property, of term-by-term integrability, gives a definition of the integral much wider and with more stable computational properties in the limit than Riemann’s integral. For example, if the approximation to a discontinuous function f(s) approaches f(x), as the number of terms of the approximations approaches infinity, then the integral of this series approximates the integral of the function in the limit; in general, this property of uniform-convergence is not true for Riemann’s integration.

As noted in Thomas Hawkins’s Lebesgue’s Theory of Integration (1970), much of the power of Lebesgue’s integration results from judicious use of the techniques of monotonic sequences and bracketing. Substituting equivalent monotonic sequences for complicated functions simplifies convergence in the limit. The bracketing technique consists of using the integrals of two well-behaved (“tame”) functions to bracket as upper and lower bounds the integral of the pathological function. Instead of subdividing the domain of the independent variable x (abscissa axis), Lebesgue subdivided the range (ordinate axis) of the corresponding function f(x) into subintervals. Therefore, Lebesgue’s integration replaces Riemann’s integral sums in the limit as sampling intervals approach 0. Lebesgue’s theory of the integral also yields other important results, such as extending the fundamental theorem of calculus.

Significance

Initially, Lebesgue’s work met with strong and lasting controversy from Borel. The main point of contention was not so much the mathematical results as the metamathematical methods used by each. Lebesgue subsequently developed the ideas of his dissertation further, and soon published these in his two classic texts, Leçons sur les series trigonométriques (1906; lessons on the trigonometric series) and Leçons sur l’integration et la recherche des fonctions primitives (1904; lessons on integration and analysis of primitive functions). Despite the fact that it was recognized early by some as an important innovation, Lebesgue’s integration was comparatively slow to be adopted by the mathematical community at large. In 1906, Lebesgue’s contemporary, the English mathematician William Henry Young, independently arrived at a somewhat more general but operationally equivalent definition of Lebesgue-type integration, using the method of monotone sequences. Most textbook discussions of Lebesgue integration have incorporated a combination of Young’s notation and formalism with Lebesgue’s arguments and examples.

Lebesgue’s integral, despite its major advantages, did not generalize completely the concept of integration for all discontinuous functions. For example, Lebesgue integration did not treat the case of unbounded functions and intervals. Subsequently, Arnaud Denjoy in 1912, Thomas Stieltjes in 1913, and Johann Radon and Maurice-René Fréchet in 1915 created other more encompassing definitions of the definite integral over complicated functions. Fréchet, in particular, showed how to generalize Lebesgue’s integral to treat functions defined on an arbitrary set without any reference to topological or metric concepts of measure, later leading to Hausdorff dimensional or (Mandelbrot) fractal measures. As further reformulated by Beppo Levi, Lebesgue integrable functions are those that almost always equal the sum of a series of step functions.

Many of the complicated functions of aerodynamics and fluid dynamics, electromagnetic theory, and the theory of probability were for the first time analytically integrable using Lebesgue’s method. In his book on the axiomatic foundations of Andrey Markov’s probability theory, A. N. Kolmogorov defined a number of operational analogues between the Borel-Lebesgue measure of a set and the probability of an event.

Bibliography

Bear, H. S. A Primer of Lebesgue Integration. 2d ed. San Diego, Calif.: Academic Press, 2002. Explains the principles and importance of Lebesgue’s contribution to the calculus.

Craven, B. O. The Lebesgue Measure and Integral. Boston: Pitman Press, 1981. An intermediate treatment, but more comprehensive. Includes a modern presentation of Borel’s set measure.

Kestelman, Hyman. Modern Theories of Integration. Oxford, England: Clarendon Press, 1937. Gives a complete but rather abstract synopsis of contemporary integration theory.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1990. The standard reference for the history of limit and function theory.

Monna, A. F. “The Integral from Riemann to Bourbaki.” In Sets and Integration, edited by Dirk Van Dalen and A. F. Monna. Groningen, the Netherlands: Wolters-Noordhoff, 1972. Provides a more technical discussion.

Temple, George E J. The Structure of Lebesgue Integration Theory. Oxford, England: Clarendon Press, 1971. The most detailed, step-by-step treatment of the modern integration theory.

Young, W. H., and G. C. Young. The Theory of Sets of Points. Cambridge, England: Cambridge University Press, 1906. Contains Young’s alternative independent development of what is essentially Lebesgue measure-based integration theory.