A Course on Integration Theory: including more than 150 exercises with detailed answers

by Nicolas Lerner

Overview

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.​

Product Details

ISBN-13:	9783034806930
Publisher:	Springer Basel
Publication date:	02/25/2014
Edition description:	2014
Pages:	492
Product dimensions:	6.10(w) x 9.25(h) x 0.04(d)

About the Author

Nicolas Lerner is Professor at Université Pierre and Marie Curie in Paris, France. He held professorial positions in the United States (Purdue University), and in France. His research work is concerned with microlocal analysis and partial differential equations. His recent book Metrics on the Phase Space and Non-Selfadjoint Pseudodifferential Operators was published by Birkhäuser. He was an invited section speaker at the Beijing International Congress of Mathematicians in 2002.

Table of Contents

1 Introduction.- 2 General theory of integration.- 3 Construction of the Lebesgue measure on Rsubd.- 4 Spaces of integrable functions.- 5 Integration on a product space.- 6 Diffeomorphisms of open subsets of Rsubd and integration.- 7 Convolution.- 8 Complex measures.- 9 Harmonic analysis.- 10 Classical inequalities.

Editorial Reviews

It is well written and the proofs are given in great detail, so that it can serve as a textbook for students as well as a reference for more advanced readers. It consists of nine chapters and an appendix devoted to making the book as self-contained as possible.” (José Rodríguez, Mathematical Reviews, October, 2016)

From the Publisher