Editing Henri Lebesgue (section)

==Lebesgue's theory of integration==
[[File:Riemann.gif|thumb|Approximation of the Riemann integral by rectangular areas]]

{{hatnote|This is a historical overview. For a technical mathematical treatment, see ''[[Lebesgue integration]]''.}}

[[integral|Integration]] is a mathematical operation that corresponds to the informal idea of finding the [[area]] under the [[graph (function)|graph]] of a [[function (mathematics)|function]]. The first theory of integration was developed by [[Archimedes]] in the 3rd century BC with his method of [[Quadrature (geometry)|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 Leibniz|Gottfried Wilhelm Leibniz]] discovered the idea that integration was intrinsically linked to [[derivative|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 of a function|limit]] and the closely related notion of [[Limit of a sequence|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 [[Infinitesimal|infinitesimally small numbers]], today rejected in standard [[mathematical analysis]]. Before Cauchy, [[Bernard Bolzano]] had laid the fundamental groundwork of the epsilon-delta definition. See [[Limit of a function#History|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 [[rectangle]]s and takes the limit of the [[summation|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 (function)|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 function]]s; measurable functions that take only [[finite set|finitely]] many values. Then he defined it for more complicated functions as the [[supremum|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.

As part of the development of Lebesgue integration, Lebesgue invented the concept of [[Lebesgue measure|measure]], which extends the idea of [[length]] from intervals to a very large class of sets, called measurable sets (so, more precisely, [[simple function]]s are functions that take a finite number of values, and each value is taken on a measurable set).
Lebesgue's technique for turning a [[measure (mathematics)|measure]] into an integral generalises easily to many other situations, 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, [[manifold]]s), 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.<ref>Weiner, N., ''[[Cybernetics: Or Control and Communication in the Animal and the Machine]]'' pp47-56 </ref> The notions of ''average'' and ''measure'' were urgently needed to provide a rigorous proof of Gibbs' [[ergodic hypothesis]].<ref>Weiner, N., ''The Fourier Integral and Certain of its Applications.''</ref>