Description 
1 online resource 
Contents 
1 Introduction  2 General theory of integration  3 Construction of the Lebesgue measure on R̂d  4 Spaces of integrable functions  5 Integration on a product space  6 Diffeomorphisms of open subsets of R̂d and integration  7 Convolution  8 Complex measures  9 Harmonic analysis  10 Classical inequalities 
Summary 
This textbook provides a detailed treatment of abstract integration theory, construction of the Lebesgue measure via the RieszMarkov Theorem and also via the Carathodory 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, changeofvariables 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 HardyLittlewoodSobolev inequality, are proven. Further topics include the RadonNikodym 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 
Bibliography 
Includes bibliographical references 
Notes 
Print version record 
Subject 
Measure theory.


Integrals, Generalized.


Measure theory  Problems, exercises, etc.


Integrals, Generalized  Problems, exercises, etc.

Genre/Form 
Problems and exercises.


Problems and exercises.

Form 
Electronic book

ISBN 
3034806949 (electronic bk.) 

9783034806947 (electronic bk.) 
