"The author aims to present 'a straightforward treatment of the part of measure theory necessary for analysis and probability' assuming only basic knowledge of analysis and topology...Each chapter includes numerous well-chosen exercises, varying from very routine practice problems to important extensions and developments of the theory; for the difficult ones there are helpful hints. It is the reviewer's opinion that the author has succeeded in his aim. In spite of its lack of new results, the selection and presentation of materials makes this a useful book for an introduction to measure and integration theory." —Mathematical Reviews "The book is a comprehensive and clearly written textbook on measure and integration...The book contains appendices on set theory, algebra, calculus and topology in Euclidean spaces, topological and metric spaces, and the Bochner integral. Each section of the book contains a number of exercises." —Zentralblatt MATH
Author(s): Cohn D.
Publisher: Birkhauser
Year: 1980
Language: English
Pages: 384
Tags: Математика;Функциональный анализ;
Contents......Page f003.djvu
Preface......Page f007.djvu
1. Algebras and sigma-algebras......Page p001.djvu
2. Measures......Page p008.djvu
3. Outer measures......Page p014.djvu
4. Lebesgue measure......Page p026.djvu
5. Completeness and regularity......Page p035.djvu
6. Dynkin classes......Page p044.djvu
1. Measurable functions......Page p048.djvu
2. Properties that hold almost everywhere......Page p058.djvu
3. The integral......Page p061.djvu
4. Limit theorems......Page p070.djvu
5. The Riemann integral......Page p075.djvu
6. Measurable functions again, complex-valued functions, and image measures......Page p079.djvu
1. Modes of convergence......Page p085.djvu
2. Normed spaces......Page p090.djvu
3. Definition of ℒ^p and L^p......Page p098.djvu
4. Properties of ℒ^p and L^p......Page p106.djvu
5. Dual spaces......Page p113.djvu
1. Signed and complex measures......Page p121.djvu
2. Absolute continuity......Page p131.djvu
3. Singularity......Page p140.djvu
4. Functions of bounded variation......Page p143.djvu
5. The duals of the L^p spaces......Page p149.djvu
1. Constructions......Page p154.djvu
2. Fubini's theorem......Page p158.djvu
3. Applications......Page p162.djvu
1. Change of variable in R^d......Page p167.djvu
2. Differentiation of measures......Page p177.djvu
3. Differentiation of functions......Page p184.djvu
1. Locally compact spaces......Page p196.djvu
2. The Riesz representation theorem......Page p205.djvu
3. Signed and complex measures; duality......Page p217.djvu
4. Additional properties of regular measures......Page p226.djvu
5. The μ^*-measurable sets and the dual of L^1......Page p232.djvu
6. Products of locally compact spaces......Page p240.djvu
1. Polish spaces......Page p251.djvu
2. Analytic sets......Page p261.djvu
3. The separation theorem and its consequences......Page p272.djvu
4. The measurability of analytic sets......Page p278.djvu
5. Cross sections......Page p284.djvu
6. Standard, analytic, Lusin, and Souslin spaces......Page p288.djvu
1. Topological groups......Page p297.djvu
2. The existence and uniqueness of Haar measure......Page p303.djvu
3. Properties of Haar measure......Page p312.djvu
4. The algebras L^1(G) and M(G)......Page p317.djvu
A. Notation and set theory......Page p328.djvu
B. Algebra......Page p334.djvu
C. Calculus and topology in R^d......Page p339.djvu
D. Topological spaces and metric spaces......Page p342.djvu
E. The Bochner integral......Page p350.djvu
Bibliography......Page p361.djvu
Index of notation......Page p367.djvu
Index......Page p369.djvu
