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A course on integration theory: including more than 150 exercises with detailed answers

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 integrab...

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Detalles Bibliográficos
Autor principal: Lerner, Nicolas
Lenguaje:eng
Publicado: Springer 2014
Materias:
Acceso en línea:https://dx.doi.org/10.1007/978-3-0348-0694-7
http://cds.cern.ch/record/1666224
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author Lerner, Nicolas
author_facet Lerner, Nicolas
author_sort Lerner, Nicolas
collection CERN
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. Further topics include the Radon-Nikodym 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.
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spelling cern-16662242021-04-21T21:15:57Zdoi:10.1007/978-3-0348-0694-7http://cds.cern.ch/record/1666224engLerner, NicolasA course on integration theory: including more than 150 exercises with detailed answersMathematical Physics and MathematicsThis 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. Further topics include the Radon-Nikodym 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.Springeroai:cds.cern.ch:16662242014
spellingShingle Mathematical Physics and Mathematics
Lerner, Nicolas
A course on integration theory: including more than 150 exercises with detailed answers
title A course on integration theory: including more than 150 exercises with detailed answers
title_full A course on integration theory: including more than 150 exercises with detailed answers
title_fullStr A course on integration theory: including more than 150 exercises with detailed answers
title_full_unstemmed A course on integration theory: including more than 150 exercises with detailed answers
title_short A course on integration theory: including more than 150 exercises with detailed answers
title_sort course on integration theory: including more than 150 exercises with detailed answers
topic Mathematical Physics and Mathematics
url https://dx.doi.org/10.1007/978-3-0348-0694-7
http://cds.cern.ch/record/1666224
work_keys_str_mv AT lernernicolas acourseonintegrationtheoryincludingmorethan150exerciseswithdetailedanswers
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