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## Über dieses Buch

Responses from colleagues and students concerning the first edition indicate that the text still answers a pedagogical need which is not addressed by other texts. There are no major changes in this edition. Several proofs have been tightened, and the exposition has been modified in minor ways for improved clarity. As before, the strength of the text lies in presenting the student with the difficulties which led to the development of the theory and, whenever possi­ ble, giving the student the tools to overcome those difficulties for himself or herself. Another proverb: Give me a fish, I eat for a day. Teach me to fish, I eat for a lifetime. Soo Bong Chae March 1994 Preface to the First Edition This book was developed from lectures in a course at New College and should be accessible to advanced undergraduate and beginning graduate students. The prerequisites are an understanding of introductory calculus and the ability to comprehend "e-I) arguments. " The study of abstract measure and integration theory has been in vogue for more than two decades in American universities since the publication of Measure Theory by P. R. Halmos (1950). There are, however, very few ele­ mentary texts from which the interested reader with a calculus background can learn the underlying theory in a form that immediately lends itself to an understanding of the subject. This book is meant to be on a level between calculus and abstract integration theory for students of mathematics and physics.

## Inhaltsverzeichnis

### Chapter Zero. Preliminaries

Abstract
The purpose of this chapter is not to serve as a text on set theory, the real number system, and topology, but to indicate to the beginner exactly which concepts and results to familiarize oneself with before studying Lebesgue integration. To save the reader unnecessary effort, we shall develop most of the topics at as elementary a level as possible.
Soo Bong Chae

### Chapter I. The Riemann Integral

Abstract
In this chapter we study elementary integration theory for functions defined on closed intervals. Although we expect that the reader has had experience with integral calculus and that the ideas are familiar, we shall not require any special results to be known. For pedagogical reasons we shall first treat the Cauchy integral. After this has been done, we will study in §3 the Riemann integral. Our attention here is focused exclusively on the definition and existence, since these concepts are often mysterious even to students who have ample knowledge of the numerous applications and techniques of evaluating Riemann integrals from their study of calculus.
Soo Bong Chae

### Chapter II. The Lebesgue Integral: Riesz Method

Abstract
Soon after Riemann’s definition of the integral in 1854, its limitations became apparent. Numerous definitions of the integral for bounded as well as unbounded functions were successively proposed after 1854. At the beginning of this century, the French mathematician Henri Lebesgue (1875–1941) introduced in his doctoral dissertation at the Sorbonne, “Intégral, longueur, aire” (1902), a notion of the integral that was to become the keystone of modern analysis.
Soo Bong Chae

### Chapter III. Lebesgue Measure

Abstract
The Lebesgue theory originally was based on an improvement and generalization of the work of Emil Borel, Leçons sur la Théorie des Fonctions (1895). Borel had already presented a theory of measure for the class of sets now known as Borel sets.
Soo Bong Chae

### Chapter IV. Generalizations

Abstract
We now undertake the task of generalizing the results of Chapters II and III, which relate to the case of a closed interval [a, b], to the case of more general sets. We could have made this generalization from the beginning, but it is the author’s experience that small doses of abstraction step by step are pedagogically more sound than one full strength dose.
Soo Bong Chae

### Chapter V. Differentiation and the Fundamental Theorem of Calculus

Abstract
If f, we call the function
$$F(x) = \int_a^x {f(t)dt}$$
(1)
an indefinite integral of f if any constant c is added to the right side of (1) the result
$$F(x) = \int_a^x {f(t)dt} + c$$
is also called an indefinite integral.
Soo Bong Chae

### Chapter VI. The L p Spaces and the Riesz–Fischer Theorem

Abstract
We now depart from the study of functions to the study of function spaces. So far our interest has been in developing the Lebesgue integral. The purpose of this new chapter is to relate the Lebesgue theory of integration to functional analysis. The theory of integration developed in this book enables us to introduce certain spaces of functions that have properties which are of great importance in analysis as well as mathematical physics, in particular, quantum mechanics. These are the so-called L p spaces of measurable functions f such that |f| p is integrable. Aside from the intrinsic importance of these spaces, we also examine some applications of results in the previous chapters. One of the most important applications is to Fourier theory. As we remarked before, Fourier theory was a key motivation of the new theory of integration. We will present here the L2 version of Fourier series, and in particular establish the Riesz-Fischer theorem which identifies the L2 and l2 spaces through Fourier series. We hope that this chapter will whet the reader’s appetite for further study of abstract spaces such as Banach and Hilbert spaces.
Soo Bong Chae

### Backmatter

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