Abstract
In modern mathematics the process of computing areas and volumes is called integration. The computation of areas of geometrical figures originated almost 2,500 years ago with the introduction by Greek mathematicians of the celebrated “method of exhaustion.” This method also introduced the modern concept of limit. In the method of exhaustion, a convex figure is approximated by inscribed (or circumscribed) polygons whose areas can be calculated—and then the number of vertexes of the inscribed polygons is increased until the convex region has been “exhausted.” That is, the area of the convex region is computed as the limit of the areas of the inscribed polygons. Archimedes (287–212 B.C.) used the method of exhaustion to calculate the area of a circle and the volume of a sphere, as well as the areas and volumes of several other geometrical figures and solids. The method of exhaustion is, in fact, at the heart of all modern integration techniques.
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This terminology is useful, but a little bit eccentric. Many authors reserve the term “step function” for a simple function whose domain is a closed interval of the real line and has a representation in terms of indicators of intervals. It is handy though to have a term to indicate a simple function that is nonzero on a set of finite measure.
Unfortunately the term “Lebesgue integral” is also used to mean the (Lebesgue) integral of a function on the real line with respect to Lebesgue measure. Some authors, e.g., [103, p. 106], mean only that. It would be less ambiguous to call our Lebesgue integral an “abstract Lebesgue integral,” but we stick with our terminology. The Lebesgue integral in its general form was introduced by H. Lebesgue [155]. The present formulation of the Lebesgue integral is essentially due to P. J. Daniell [59].
The Dunford integral was introduced by N. Dunford [76] and the Pettis integral by B. J. Pettis [190].
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© 1994 Springer-Verlag Berlin Heidelberg
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Aliprantis, C.D., Border, K.C. (1994). Integrals. In: Infinite Dimensional Analysis. Studies in Economic Theory, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03004-2_9
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DOI: https://doi.org/10.1007/978-3-662-03004-2_9
Publisher Name: Springer, Berlin, Heidelberg
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