Integration is a mathematical operation that corresponds to the informal idea of finding the
area under the
graph of a
function. The first theory of integration was developed by
Archimedes in the 3rd century BC with his method of
quadratures, but this could be applied only in limited circumstances with a high degree of geometric symmetry. In the 17th century,
Isaac Newton and
Gottfried Wilhelm Leibniz discovered the idea that integration was intrinsically linked to
differentiation, the latter being a way of measuring how quickly a function changed at any given point on the graph. This surprising relationship between two major geometric operations in calculus, differentiation and integration, is now known as the
fundamental theorem of calculus. It has allowed mathematicians to calculate a broad class of integrals for the first time. However, unlike Archimedes' method, which was based on
Euclidean geometry, mathematicians felt that Newton's and Leibniz's
integral calculus did not have a rigorous foundation. The mathematical notion of
limit and the closely related notion of
convergence are central to any modern definition of integration. In the 19th century,
Karl Weierstrass developed the rigorous epsilon-delta definition of a limit, which is still accepted and used by mathematicians today. He built on previous but non-rigorous work by
Augustin Cauchy, who had used the non-standard notion of
infinitesimally small numbers, today rejected in standard
mathematical analysis. Before Cauchy,
Bernard Bolzano had laid the fundamental groundwork of the epsilon-delta definition. See
here for more.
Bernhard Riemann followed up on this by formalizing what is now called the
Riemann integral. To define this integral, one fills the area under the graph with smaller and smaller
rectangles and takes the limit of the
sums of the areas of the rectangles at each stage. For some functions, however, the total area of these rectangles does not approach a single number. Thus, they have no Riemann integral. Lebesgue invented a new method of integration to solve this problem. Instead of using the areas of rectangles, which put the focus on the
domain of the function, Lebesgue looked at the
codomain of the function for his fundamental unit of area. Lebesgue's idea was to first define measure, for both sets and functions on those sets. He then proceeded to build the integral for what he called
simple functions; measurable functions that take only
finitely many values. Then he defined it for more complicated functions as the
least upper bound of all the integrals of simple functions smaller than the function in question. Lebesgue integration has the property that every function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree. Furthermore, every bounded function on a closed bounded interval has a Lebesgue integral and there are many functions with a Lebesgue integral that have no Riemann integral. All of these advantages over Riemann integration made Lebesgue's method a more powerful tool for mathematicians. As part of the development of Lebesgue integration, Lebesgue invented the concept of
measure, which extends the idea of
length from intervals to a very large class of sets, called measurable sets. This larger class of sets meant that Lebesgue's technique for turning a
measure into an integral generalises easily to many other situations with measurable sets, leading to the modern field of
measure theory. The Lebesgue integral is deficient in one respect. The Riemann integral generalises to the
improper Riemann integral to measure functions whose domain of definition is not a
closed interval. The Lebesgue integral integrates many of these functions (always reproducing the same answer when it does), but not all of them. For functions on the real line, the
Henstock integral is an even more general notion of integral (based on Riemann's theory rather than Lebesgue's) that subsumes both Lebesgue integration and improper Riemann integration. However, the Henstock integral depends on specific ordering features of the
real line and so does not generalise to allow integration in more general spaces (say,
manifolds), while the Lebesgue integral extends to such spaces quite naturally.
Implications for statistical mechanics In 1947
Norbert Wiener claimed that the Lebesgue integral had unexpected but important implications in establishing the validity of
Willard Gibbs' work on the foundations of
statistical mechanics. The notions of
average and
measure were urgently needed to provide a rigorous proof of Gibbs'
ergodic hypothesis. ==See also==