Riemann-Stieltjes and Lebesgue Integrals
Abstract
The idea of integration has been around since the time of Archimedes,
but it has been said that "the theory of integration was a creation of
the twentieth century." In 1854 Georg F.B. Riemann gave a set of necessary
and sufficient conditions under which a bounded function is said
to be integrable. Today, a function of this type is known as being
"Riemann integrable" and almost every undergraduate student who has taken
a calculus course has learned about this form of integration.
Riemann dominated the field of integration until 1894 when a Dutch
mathematician named Thoman Jan Stieltjes developed the Riemann-
Stieltjes integral while investigating a very specific problem concerning
a thin rod of nonuniformly distibuted mass. This specific problem called for
the development of the first generalized form of the Riemann integral.
Shortly after this, Henri Lebesge developed a generalization of this generalization
which soon dominated studies in measure theory and integration and remain prominant
ideas of mathematical analysis today.
Table of Contents
Complete List of References
- [AN1]
Artemiadis, Nicolaos.
History of Mathematics: From a Mathematician's Vantage Point.
Translation from the Greek by Nikolaos E. Sofronidis, 2004.
- [BC1]
Boyer, Carl.
A History of Mathematics.
New York: John Wiley & Sons, Inc, 1968.
- [CN1]
Carothers, N.L.
Real Analysis.
New York: Cambridge University Press, 2000.
- [RM1]
Rosenlicht, Maxwell.
Introduction to Analysis.
New York: Dover Publications, 1968.
- [RW1]
Rudin, Walter.
Principles of Mathematical Analysis.
New York: McGraw-Hill, Inc., 1964.
- [LW1]
Lederman, Walter and Vajda, Steven.
Handbook of Applicable Mathematics, Volume IV: Analysis.
New York: John Wiley & Sons, 1982. pp. 160-163.
