Introduction to Integral Calculus

Fülszöveg

"This book is a short introduction to the integral calculus of real-real functions. The treatment follows my university lectures. My fundamental objective was to give an introduction of the concept of the integral that is not only simple but can also be generalised considerably. Therefore I broke with the Riemann integral which requires rather complicated constructions. First I define the integral itself for step functions and then extend it to functions which can be represented as limit functions of uniformly convergent sequences of step functions. These so-called simple functions appear in most practical applications of the integral. The idea of the construction is based on the fundamental works by F. Riesz and J. Dieudonné [F. Riesz and B. Sz.-Nagy, Functional Analysis (English edition), New York, Ungar, 1956; J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960]. The construction used in the present book admits a far-reaching generalisation: If uniform convergence is replaced by almost everywhere convergence, we obtain the construction of the Lebesgue integral, as it was given by Riesz. For the understanding of the present book familiarity with the basics of differential calculus is sufficient. The book does not contain exercises, a great number of these can be found in any textbook dealing with the elements of integral calculus. In the last chapter, however, we present some problems of theoretical and practical importance." (András Kósa)