THEORY OF THE

Brian S. Thomson Simon Fraser University

CLASSICALREALANALYSIS.COM This text is intended as a treatise for a rigorous course introducing the ele- ments of integration theory on the . All of the important features of the , the Lebesgue integral, and the Henstock-Kurzweil integral are covered. The text can be considered a sequel to the four chapters of the more elementary text THE INTEGRAL which can be downloaded from our web site. For advanced readers, however, the text is self-contained. For further information on this title and others in the series visit our website.

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Citation: Theory of the Integral, Brian S. Thomson, ClassicalRealAnalysis.com (2012), [ISBN 1467998168]

Date PDF file compiled: December 27, 2011

ISBN-13: 978-1467998161 ISBN-10: 1467998168

CLASSICALREALANALYSIS.COM Preface

Work in Progress: This is a preliminary version of a text planned for an April 2012 publication date. This file is dated December 27, 2011. Return to our website occa- sionally for updates. Please send comments to the author.

The text is a self-contained account of integration theory on the real line. The usual curricula in real analysis courses do not allow for much time to be spent on the Henstock-Kurzweil integral. Instead extensive accounts of Riemann’s integral and the Lebesgue integral are presented. Accordingly the version here would be mostly recommended for supplementary reading. Even so it would be a reasonable course design to teach this material prior to a course in abstract measure and integration. The student should end up as well-prepared as in more traditional courses. Certainly every professional math- ematician should be aware of more than Lebesgue’s theory; while nonabsolutely convergent do not play an extensive role in applications, they are part of our history and of our culture. The reader might want to view first the prequel to this text:

B. S. Thomson, The Calculus Integral, ClassicalRealAnalysis.com (2008). ISBN-13: 978-1442180956, ISBN-10: 1442180951

That text is an (experimental) outline of an elementary real analysis course in which the Newton integral plays the key role. Since the presentation in the present textbook also uses the Newton integral (in its various versions) as a motivating tool, the reader may wish also to consult the prequel to see how this would work at an introductory level. There are innumerable books published on the Lebesgue integral. The reader needing instruction in that theory is faced with too many choices, although many of them are truly excellent. For the more general integral (called here the gen- eral Newton integral or simply “the integral”) that is best known classically as the Denjoy-Perron integral and, more recently, as the Henstock-Kurzweil integral, there are far fewer choices and not all of them are excellent. I have resisted for many years writing a lengthy account of this integral, partly because the topic is not widely thought of as being of much significance. ii

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