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2014 | Book

General Theory of Integration Read chapter Read first chapter

A Course on Integration Theory

including more than 150 exercises with detailed answers

Author: Nicolas Lerner

Publisher: Springer Basel

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

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. The Radon-Nikodym theorem, notions of harmonic analysis, classical inequalities and interpolation theorems, including Marcinkiewicz's theorem, the definition of Lebesgue points and Lebesgue differentiation theorem are further topics included. A detailed appendix provides the reader with various elements of elementary mathematics, such as a discussion around the calculation of antiderivatives or the Gamma function. The appendix also provides more advanced material such as some basic properties of cardinals and ordinals which are useful in the study of measurability.​

Table of Contents

Frontmatter
Chapter 1. General Theory of Integration
Nicolas Lerner
Chapter 2. Actual Construction of Measure Spaces
Nicolas Lerner
Chapter 3. Spaces of Integrable Functions
Nicolas Lerner
Chapter 4. Integration on a Product Space
Nicolas Lerner
Chapter 5. Diffeomorphisms of Open Subsets of ℝ n and Integration
Nicolas Lerner
Chapter 6. Convolution
Nicolas Lerner
Chapter 7. Complex Measures
Nicolas Lerner
Chapter 8. Basic Harmonic Analysis on ℝ n
Nicolas Lerner
Chapter 9. Classical Inequalities
Nicolas Lerner
Chapter 10. Appendix
Nicolas Lerner
Backmatter
Metadata
Title
A Course on Integration Theory
Author
Nicolas Lerner
Copyright Year
2014
Publisher
Springer Basel
Electronic ISBN
978-3-0348-0694-7
Print ISBN
978-3-0348-0693-0
DOI
https://doi.org/10.1007/978-3-0348-0694-7

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