This textbook provides a detailed treatment of abstract integration theory, construction of the Lebesgue measure via the Riesz-Markov Theorem and also via the Caratheodory 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.

About the Author
Nicolas Lerner is Professor at Universite 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 Birkhauser. He was an invited section speaker at the Beijing International Congress of Mathematicians in 2002.

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." (Jose Rodriguez, Mathematical Reviews, October, 2016)

Book Information
ISBN 9783034806930
Author Nicolas Lerner
Format Paperback
Page Count 492
Imprint Birkhauser Verlag AG
Publisher Birkhauser Verlag AG