A Course on Integration Theory
including more than 150 exercises with detailed answers
Published on 9. July 2014
XVIII, 492 pages
978-3-0348-0694-7 (ISBN)
Description
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.
Reviews / Votes
"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)More details
Series
Language
English
Place of publication
Basel
Switzerland
Publishing group
Springer Basel
Target group
Primary & secondary/elementary & high school
Illustrations
XVIII, 492 p. 15 illus., 3 illus. in color.
File size
5,73 MB
ISBN-13
978-3-0348-0694-7 (9783034806947)
DOI
10.1007/978-3-0348-0694-7
Schweitzer Classification
Other editions
Additional editions
Nicolas Lerner
A Course on Integration Theory
including more than 150 exercises with detailed answers
Book
03/2014
Birkhäuser
€85.59
Shipment within 10-15 days
Person
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.
Content
1 Introduction.- 2 General theory of integration.- 3 Construction of the Lebesgue measure on R^d.- 4 Spaces of integrable functions.- 5 Integration on a product space.- 6 Diffeomorphisms of open subsets of R^d and integration.- 7 Convolution.- 8 Complex measures.- 9 Harmonic analysis.- 10 Classical inequalities.
