Measure Theory, Volume 1Springer Science & Business Media, 15 janv. 2007 - 1075 pages Measure theory is a classical area of mathematics born more than two thousand years ago. Nowadays it continues intensive development and has fruitful connections with most other fields of mathematics as well as important applications in physics. This book gives an exposition of the foundations of modern measure theory and offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course (the material of this level corresponds to a variety of special courses), and, finally, more specialized topics partly covered by more than 850 exercises. Volume 1 (Chapters 1-5) is devoted to the classical theory of measure and integral. Whereas the first volume presents the ideas that go back mainly to Lebesgue, the second volume (Chapters 6-10) is to a large extent the result of the later development up to the recent years. The central subjects in Volume 2 are: transformations of measures, conditional measures, and weak convergence of measures. These three topics are closely interwoven and form the heart of modern measure theory. The organization of the book does not require systematic reading from beginning to end; in particular, almost all sections in the supplements are independent of each other and are directly linked only to specific sections of the main part. The target readership includes graduate students interested in deeper knowledge of measure theory, instructors of courses in measure and integration theory, and researchers in all fields of mathematics. The book may serve as a source for many advanced courses or as a reference. |
Table des matières
1 | |
not countably additive measures 67 Abstract inner measures 70 | 70 |
The Lebesgue integral | 105 |
on IRn 145 The functional monotone class theorem 146 Baire | 156 |
Operations on measures and functions | 175 |
The spaces Lp and spaces of measures | 249 |
Hellingers distance 299 Additive set functions 302 Exercises 303 | 303 |
Connections between the integral and derivative | 329 |
and Fanalytic sets 49 Blackwell spaces 50 Mappings of Souslin | 54 |
Measures on topological spaces 67 | 67 |
Weak convergence of measure 175 | 175 |
Transformations of measures and isomorphisms 267 | 267 |
expectations 339 | 339 |
Bibliographical and Historical | 439 |
References 465 | 465 |
512 | |
Preface to Volume 2 v | 504 |
Conditional measures and conditional | 508 |
Borel Baire and Souslin sets 1 | 1 |
547 | |
561 | |
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Références à ce livre
Limit Theorems for Associated Random Fields and Related Systems Aleksandr Vadimovich Bulinskiĭ,Alexey Shashkin Aucun aperçu disponible - 2007 |