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© 2013

Measure, Integral, Derivative

A Course on Lebesgue's Theory

  • Contains ?all the main results of Lebesgue’s theory

  • Accessible to both upper-undergraduate and master’s students

  • Includes 180+ exercises with varying degrees of difficulty

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-X
  2. Sergei Ovchinnikov
    Pages 1-26
  3. Sergei Ovchinnikov
    Pages 27-64
  4. Sergei Ovchinnikov
    Pages 65-95
  5. Sergei Ovchinnikov
    Pages 97-127
  6. Back Matter
    Pages 129-146

About this book

Introduction

This classroom-tested text is intended for a one-semester course in Lebesgue’s theory.  With over 180 exercises, the text takes an elementary approach, making it easily accessible to both upper-undergraduate- and lower-graduate-level students.  The three main topics presented are measure, integration, and differentiation, and the only prerequisite is a course in elementary real analysis.

In order to keep the book self-contained, an introductory chapter is included with the intent to fill the gap between what the student may have learned before and what is required to fully understand the consequent text. Proofs of difficult results, such as the differentiability property of functions of bounded variations, are dissected into small steps in order to be accessible to students. With the exception of a few simple statements, all results are proven in the text.  The presentation is elementary, where σ-algebras are not used in the text on measure theory and Dini’s derivatives are not used in the chapter on differentiation. However, all the main results of Lebesgue’s theory are found in the book.

Keywords

Lebesgue integration Lebesgue measure analysis differentiability of BV-functions measurable sets

Authors and affiliations

  1. 1.Dept. MathematicsSan Francisco State UniversitySan FranciscoUSA

About the authors

Sergei Ovchinnikov is currently Professor of Mathematics at San Francisco State University.

Bibliographic information

  • Book Title Measure, Integral, Derivative
  • Book Subtitle A Course on Lebesgue's Theory
  • Authors Sergei Ovchinnikov
  • Series Title Universitext
  • Series Abbreviated Title Universitext
  • DOI https://doi.org/10.1007/978-1-4614-7196-7
  • Copyright Information Springer Science+Business Media New York 2013
  • Publisher Name Springer, New York, NY
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-1-4614-7195-0
  • eBook ISBN 978-1-4614-7196-7
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages X, 146
  • Number of Illustrations 16 b/w illustrations, 0 illustrations in colour
  • Topics Measure and Integration
    Real Functions
    Analysis
  • Buy this book on publisher's site

Reviews

From the reviews:

“It is accessible to upper-undergraduate and lower graduate level students, and the only prerequisite is a course in elementary real analysis. … The book proposes 187 exercises where almost always the reader is proposed to prove a statement. … this book is a very helpful tool to get into Lebesgue’s theory in an easy manner.” (Daniel Cárdenas-Morales, zbMATH, Vol. 1277, 2014)

“This is a brief … but enjoyable book on Lebesgue measure and Lebesgue integration at the advanced undergraduate level. … The presentation is clear, and detailed proofs of all results are given. … The book is certainly well suited for a one-semester undergraduate course in Lebesgue measure and Lebesgue integration. In addition, the long list of exercises provides the instructor with a useful collection of homework problems. Alternatively, the book could be used for self-study by the serious undergraduate student.” (Lars Olsen, Mathematical Reviews, December, 2013)