Henri Lebesgue - Lebesgue's Theory of Integration

Lebesgue's Theory of Integration

This is a non-technical treatment from a historical point of view; see the article Lebesgue integration for a technical treatment from a mathematical point of view.

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 independently discovered the idea that integration was roughly the inverse operation of differentiation, the latter being a way of measuring how quickly a function changed at any given point on the graph. This 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.

In the 19th century, Augustin Cauchy developed epsilon-delta limits, and 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. As such, 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 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 bounded function defined over a bounded interval with a Riemann integral also has a Lebesgue integral, and for those functions the two integrals agree. But 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 measure, which extends the idea of length from intervals to a very large class of sets, called measurable sets (so, more precisely, simple functions 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 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 did), 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.