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.
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