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# Lebesgue's Development of the Theories of Measure and Integration

## Overview

Henri Lebesgue (1875-1941) revived the troubled field of integration. His generalization of integration, and the complex theory of measure he introduced to accomplish this, countered the criticisms and challenges to the field that threatened it at the end of the nineteenth century.

## Background

Integration can be thought of in two ways. First, as the opposite of differentiation, so an integral is an anti-derivative. However, this is a very abstract concept. Second, integration between two points can be seen as the method of calculating the area of a shape where at least one side is not straight, but varies according to some function. While the calculation of the area of a square or triangle is straightforward, the area of, for example, a "D"-or "B"-shaped area is much more challenging mathematically. Often, these problems are thought of in terms of finding the area under a curve on a plotted graph, which is how they are generally presented in textbooks.

Many Greek mathematicians were concerned with such problems. Methods for calculating the areas of squares, triangles, and other regular shapes were understood, but once an object had a curve it seemed impossible to compute. Various methods were considered by thinkers such as Archimedes (287-212 b.c.). One popular method used the idea of upper and lower limits. If you take the shape you wish to measure and draw a regular polygon that would fit just inside the irregular shape, this gives a lower limit to the area of the irregular shape. Then draw a regular polygon that only just completely surrounds the irregular shape. This gives an upper limit to the shape, so the actual area must lie between the two areas drawn. If one regular shape was not close enough, then several packed together like a jigsaw could be used. While this method could get close approximations, it was in a sense a self-defeating process, as you could never get the correct answer.

Centuries later European mathematicians such as Johannes Kepler (1571-1630) introduced the use of infinitesimals to their calculations of areas. Infinitesimals were thought of as quantities smaller than any actual finite quantity, but not quite zero. While this is a somewhat strange idea, it was very influential. The first textbook on integration methods was published in 1635 by Bonaventura Cavalieri (1598-1647), but in a form quite unrecognizable today. Further modifications were made by various thinkers, each using their own special method for approximating the area under a curve. Many methods worked well for one kind of curve but poorly for others. The modern form of integration began to take shape when both Isaac Newton (1643-1727) and Gottfried von Leibniz (1646-1716) worked independently on calculus and developed a more general method for finding the area under a curve. A bitter argument ensued over who had discovered the idea first. Newton won in the short term; his anonymously written attacks on Leibniz were numerous, insulting, and vicious. However, in the long term it was the simpler style and notation of Leibniz that had a larger impact on mathematics.

Integration was further refined by mathematicians such as Augustin Cauchy (1789-1857) and Karl Weierstrass (1815-1897). The idea of infinitestimals was replaced with the concept of limits, in a sense going back to Archimedes' ideas. Cauchy gave elementary calculus the form it still holds today. It was from his concept of the integral as a limit of a sum, rather than simply an anti-derivative, that many modern concepts of integration have come.

In the middle of the nineteenth century Bernhard Riemann (1826-1866) defined integration more precisely, expanding the functions for which the method was useful. His work was in turn tinkered with over the next few decades by a number of theorists, so that it applied a little more generally.

However, towards the end of the nineteenth century many exceptions to Riemann integration became obvious. In integration and many other fields of mathematics, younger mathematicians took great delight in finding cases that broke the rules, cases that were termed "pathological examples." For Riemann integration these were often functions that had many discontinuous points, that is where a curve was not smooth and flowing but had gaps or points of extreme, instantaneous change. Another problem was the number of cases in which the inverse relationship between integration and differentiation did not appear to hold. Since one way of treating integration was as the opposite of differentiation, finding cases where this was not true suggested that something was wrong with the whole field. A number of established mathematicians feared mathematics was crumbling around them, as the whole basis of the discipline seemed under attack. Some critics called for the abandonment of integration altogether.

Henri Lebesgue was an unlikely savior of integration. He had a fine but not outstanding, university career. After completing his studies at the Ecole Normale Supérieure in Paris, France, he worked in its library for two years, during which time he read the thesis of another recent graduate, René Baire (1874-1932). Baire's doctorate, written in 1899, contained work on discontinuous functions, just the type of pathological cases that could not be solved using Riemann integration. Baire's poor health limited further work, but Lebesgue saw the potential of the ideas.

Lebesgue was also influenced by the work of a number of other mathematicians. Camille Jordan (1838-1922) had extended the Riemann integral by using the notion of measure. Emile Borel (1871-1956) radically revised what the notions of measure and measurability were in regards to sets. The mathematical idea of measure is not easily explained in words, as it is mathematically complex. It is an extension of the familiar idea of measuring areas (length, area, volume) generalized to the wider system of sets, and so deals with abstract spaces.

Borel had used radical new ideas in the field of set theory in his work, specifically the controversial work of Georg Cantor (1845-1918). Cantor's set theory was strongly opposed by some mathematicians, as it dealt with infinities in ways that contradicted some philosophical and religious doctrines. Mathematicians, and the wider intellectual community, had long been uncomfortable with the notion of infinity, and for some Cantor's argument that some infinite quantities were bigger than others seemed to go too far.

By putting all these divergent elements together Lebesgue revolutionized integration. In 1901 he introduced his theory of measure. This extended the ideas of Jordan and Borel and greatly increased the generality of the concept. The definition he presented is still called the Lebesgue measure. More importantly, however, it also had implications for integration.

## Impact

In 1902 Lebesgue used his concept of measure to broaden the scope of integration far beyond the definitions of Riemann. He realized that the Riemann integral only applied in exceptional cases and that he needed to make integration into a more robust and general concept if it was to include the ideas he had read in Baire's thesis. At a stroke Lebesgue answered most of the questions that had been asked of integration, and his work paved the way for further refinement. He showed how his new definition of the integral, called the Lebesgue integral, could now handle the pathological cases that had been used to undermine the field. The Lebesgue integral had far fewer cases where integration was not the inverse of differentiation, and it gave hope that more work would reduce these further.

Lebesgue's work helped to reassert the fundamental importance of integration and resolved a century-long discussion of the differentiability properties of continuous functions. His work on measure and integration greatly expanded the scope of Fourier analysis and is one of the foundations of modern mathematical analysis.

Lebesgue's radical ideas were presented in his thesis, submitted at the University of Nancy in 1902. At first his work attracted little attention. However, in 1904 and 1906 he published two books that helped popularize his arguments. Strong criticism of his work emerged, particularly as it relied on notions taken from Cantor's controversial set theory. Slowly, however, more and more mathematicians began to use the Lebesgue measure and the Lebesgue integral in their work. As the Lebesgue integral became more accepted, it helped boost the validity of Cantor's set theory.

By 1910 Lebesgue's work was well received enough that he was offered a position at the Sorbonne in Paris. However, Lebesgue did not concentrate on the field he had revived. Rather he moved on to other areas of mathematics, such as topology, potential theory, and Fourier analysis. He thought his work on the integral had been a striking generalization, and he was fearful of what generalizations were doing to mathematics. He said, "Reduced to general theories, mathematics would be a beautiful form without content. It would quickly die." Yet the field he created flourished in the hands of others.

Arnard Denjoy (1884-1974) extended the definition of the Lebesgue integral (the Denjoy integral), and Alfred Haar (1885-1933) did further work (the Haar integral). Lebesgue's ideas were combined with that of his contemporary T. J. Stieltjes (1856-1894) by a third mathematician to give the Lebesgue-Stieltjes integral—yet another extension of integration, allowing even more functions to be included. The work done by Lebesgue and those that followed him led some to suggest that while the concept of integration was at least as old as Archimedes, the theory of integration was a creation of the twentieth century.

Lebesgue's ideas are complex and abstract. Indeed, his ideas of measure and integration are generally only taught at post-graduate level in most universities. For the majority of cases the Riemann integral is still used, and the vast majority of undergraduate calculus courses worldwide still use Riemann's definition to explain integration. Most elementary calculus courses do not even go as far as Riemann's ideas, and the format of Cauchy's calculus is still taught today. However, while Lebesgue's ideas may be abstract and difficult to comprehend, the continued success and mathematical viability of the field of integration—at all levels—owes a great deal to his work in reviving the discipline.

DAVID TULLOCH