The Elements of Integration and Lebesgue Measure

This book consists of two separate, but closely related, parts. The first part (Chapters 1-10) is subtitled The Elements of Integration; the second part (Chapters 11-17) is subtitled The Elements of Lebesgue measure. It is possible to read these two parts in either order, with only a bit of repetition. The Elements of Integration is essentially a corrected reprint of a book with that title, originally published in 1966, designed to present the chief results of the Lebesgue theory of integration to a reader having only a modest mathematical background. This book developed from the lectures given by the author at the University of Illinois, Urbana Champaign, and it was subsequently used there and elsewhere with considerable success. Its only prerequisites are a understanding of elementary real analysis and the ability to comprehend "- S arguments". It is assumed that the reader has some familarity with the Riemann integral so that it is not necessary to provide motivation and detailed discussion, but do not assume that the reader has a mastery of the subtleties of that theory. A solid course in "advanced calculus", an understanding of the first third of the other author's books The Elements of Real Analysis, or of most of the book Introduction to Real Analysis co-authord with D. R. Sherbert provides an adequate background. In preparing this new edition the author seized the opportunity to correct certain errors, but have resisted the temptation to insert additional material, since he believe that one of the features of this book that is most appreciated is its brevity. The Elements of Lebesgue Measure is descended from class notes written to acquaint the reader with the theory of Lebesgue measure in the space RP. While it is easy to find good treatments of the case p = 1, the case p > 1 is not quite as simple and is much less frequently discussed. The main ideas of Lebesgue measure are presented in detail in Chapters 10-15, although some relatively easy remarks are left to the reader as exercises. The final two chapters venture into the topic of nonmeasurabl a sets and round out the subject. There are many expositions of the Lebesgue integral from various points of view, but I believe that the abstract measure space approach used here strikes directly towards the most important results: the convergence theorems. Further, this approach is particularly wellsuited for students of probability and statistics, as well as students of analysis. Since the book is intended as an introduction, one do not follow all of the avenues that are encountered. However, it take pains not to attain brevity by leaving out important details, or assigning them to the reader. Readers who complete this book are certainly not through, but if this book helps to speed them on their way, it has accomplished its purpose. In the References, some books that the author believe readers can profitably explore, as well as works cited in the body of the text.