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A Course on Integration Theory

including more than 150 exercises with detailed answers

  • Nicolas Lerner

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Nicolas Lerner
    Pages 1-66
  3. Nicolas Lerner
    Pages 67-123
  4. Nicolas Lerner
    Pages 125-187
  5. Nicolas Lerner
    Pages 189-217
  6. Nicolas Lerner
    Pages 283-316
  7. Nicolas Lerner
    Pages 317-341
  8. Nicolas Lerner
    Pages 343-370
  9. Nicolas Lerner
    Pages 371-406
  10. Nicolas Lerner
    Pages 407-479
  11. Back Matter
    Pages 481-492

About this book

Introduction

This textbook provides a detailed treatment of abstract integration theory, construction of the Lebesgue measure via the Riesz-Markov Theorem and also via the Carathéodory Theorem. It also includes some elementary properties of Hausdorff measures as well as the basic properties of spaces of integrable functions and standard theorems on integrals depending on a parameter. Integration on a product space, change-of-variables formulas as well as the construction and study of classical Cantor sets are treated in detail. Classical convolution inequalities, such as Young's inequality and Hardy-Littlewood-Sobolev inequality, are proven. Further topics include the Radon-Nikodym theorem, notions of harmonic analysis, classical inequalities and interpolation theorems including Marcinkiewicz's theorem, and the definition of Lebesgue points and the Lebesgue differentiation theorem.

Each chapter ends with a large number of exercises and detailed solutions.

A comprehensive appendix provides the reader with various elements of elementary mathematics, such as a discussion around the calculation of antiderivatives or the Gamma function. It also provides more advanced material such as some basic properties of cardinals and ordinals which are useful for the study of measurability.

Keywords

Fourier transformation L^p spaces Lebesgue measure measure theory

Authors and affiliations

  • Nicolas Lerner
    • 1
  1. 1.Institut de Mathématiques de JussieuUniversité Pierre et Marie Curie (Paris VI)ParisFrance

Bibliographic information

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