Advanced Integration Theory
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Advanced Integration Theory
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Author : Corneliu Constantinescu
language : en
Publisher: Springer
Release Date : 2014-09-01
Advanced Integration Theory written by Corneliu Constantinescu and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-09-01 with Mathematics categories.
Since about 1915 integration theory has consisted of two separate branches: the abstract theory required by probabilists and the theory, preferred by analysts, that combines integration and topology. As long as the underlying topological space is reasonably nice (e.g., locally compact with countable basis) the abstract theory and the topological theory yield the same results, but for more compli cated spaces the topological theory gives stronger results than those provided by the abstract theory. The possibility of resolving this split fascinated us, and it was one of the reasons for writing this book. The unification of the abstract theory and the topological theory is achieved by using new definitions in the abstract theory. The integral in this book is de fined in such a way that it coincides in the case of Radon measures on Hausdorff spaces with the usual definition in the literature. As a consequence, our integral can differ in the classical case. Our integral, however, is more inclusive. It was defined in the book "C. Constantinescu and K. Weber (in collaboration with A.
Advanced Integration Theory
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Author : Corneliu Constantinescu
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Advanced Integration Theory written by Corneliu Constantinescu and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Since about 1915 integration theory has consisted of two separate branches: the abstract theory required by probabilists and the theory, preferred by analysts, that combines integration and topology. As long as the underlying topological space is reasonably nice (e.g., locally compact with countable basis) the abstract theory and the topological theory yield the same results, but for more compli cated spaces the topological theory gives stronger results than those provided by the abstract theory. The possibility of resolving this split fascinated us, and it was one of the reasons for writing this book. The unification of the abstract theory and the topological theory is achieved by using new definitions in the abstract theory. The integral in this book is de fined in such a way that it coincides in the case of Radon measures on Hausdorff spaces with the usual definition in the literature. As a consequence, our integral can differ in the classical case. Our integral, however, is more inclusive. It was defined in the book "C. Constantinescu and K. Weber (in collaboration with A.
Measure And Integration
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Author : Heinz König
language : en
Publisher: Springer Science & Business Media
Release Date : 1997
Measure And Integration written by Heinz König and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Mathematics categories.
This book aims at restructuring some fundamentals in measure and integration theory. It centers around the ubiquitous task to produce appropriate contents and measures from more primitive data like elementary contents and elementary integrals. It develops the new approach started around 1970 by Topsoe and others into a systematic theory. The theory is much more powerful than the traditional means and has striking implications all over measure theory and beyond.
A Concise Introduction To The Theory Of Integration
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Author : Daniel W. Stroock
language : en
Publisher: Springer Science & Business Media
Release Date : 1998-12-23
A Concise Introduction To The Theory Of Integration written by Daniel W. Stroock and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1998-12-23 with Mathematics categories.
Designed for the analyst, physicist, engineer, or economist, provides such readers with most of the measure theory they will ever need. Emphasis is on the concrete aspects of the subject. Subjects include classical theory, Lebesgue's measure, Lebesgue integration, products of measures, changes of variable, some basic inequalities, and abstract theory. Annotation copyright by Book News, Inc., Portland, OR
A Modern Theory Of Integration
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Author : Robert G. Bartle
language : en
Publisher: American Mathematical Society
Release Date : 2024-10-25
A Modern Theory Of Integration written by Robert G. Bartle and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-10-25 with Mathematics categories.
The theory of integration is one of the twin pillars on which analysis is built. The first version of integration that students see is the Riemann integral. Later, graduate students learn that the Lebesgue integral is ?better? because it removes some restrictions on the integrands and the domains over which we integrate. However, there are still drawbacks to Lebesgue integration, for instance, dealing with the Fundamental Theorem of Calculus, or with ?improper? integrals. This book is an introduction to a relatively new theory of the integral (called the ?generalized Riemann integral? or the ?Henstock-Kurzweil integral?) that corrects the defects in the classical Riemann theory and both simplifies and extends the Lebesgue theory of integration. Although this integral includes that of Lebesgue, its definition is very close to the Riemann integral that is familiar to students from calculus. One virtue of the new approach is that no measure theory and virtually no topology is required. Indeed, the book includes a study of measure theory as an application of the integral. Part 1 fully develops the theory of the integral of functions defined on a compact interval. This restriction on the domain is not necessary, but it is the case of most interest and does not exhibit some of the technical problems that can impede the reader's understanding. Part 2 shows how this theory extends to functions defined on the whole real line. The theory of Lebesgue measure from the integral is then developed, and the author makes a connection with some of the traditional approaches to the Lebesgue integral. Thus, readers are given full exposure to the main classical results. The text is suitable for a first-year graduate course, although much of it can be readily mastered by advanced undergraduate students. Included are many examples and a very rich collection of exercises. There are partial solutions to approximately one-third of the exercises. A complete solutions manual is available separately.
Integration A Functional Approach
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Author : Klaus Bichteler
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-11-23
Integration A Functional Approach written by Klaus Bichteler and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-23 with Mathematics categories.
This book covers Lebesgue integration and its generalizations from Daniell's point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main purpose of measure theory. From this point of view Lebesgue's integral can be had as a rather straightforward, even simplistic, extension of Riemann's integral; and its aims, definitions, and procedures can be motivated at an elementary level. The notion of measurability, for example, is suggested by Littlewood's observations rather than being conveyed authoritatively through definitions of (sigma)-algebras and good-cut-conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner. The approach taken provides the additional benefit of cutting the labor in half. The use of seminorms, ubiquitous in modern analysis, speeds things up even further. The book is intended for the reader who has some experience with proofs, a beginning graduate student for example. It might even be useful to the advanced mathematician who is confronted with situations - such as stochastic integration - where the set-measuring approach to integration does not work.
Geometric Integration Theory
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Author : Hassler Whitney
language : en
Publisher: Courier Corporation
Release Date : 2005-12-10
Geometric Integration Theory written by Hassler Whitney and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-12-10 with Mathematics categories.
This treatment of geometric integration theory consists of an introduction to classical theory, a postulational approach to general theory, and a section on Lebesgue theory. Covers the theory of the Riemann integral; abstract integration theory; some relations between chains and functions; Lipschitz mappings; chains and additive set functions, more. 1957 edition.
Lectures On The Theory Of Integration
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Author : Ralph Henstock
language : en
Publisher: World Scientific Publishing Company Incorporated
Release Date : 1988
Lectures On The Theory Of Integration written by Ralph Henstock and has been published by World Scientific Publishing Company Incorporated this book supported file pdf, txt, epub, kindle and other format this book has been release on 1988 with Mathematics categories.
Ch. 1. Introduction. 1. The Riemann and Riemann-Darboux integrals. 2. Modifications using the mesh and refinement of partitions. 3. The calculus indefinite integral and the Riemann-complete or generalized Riemann integral -- ch. 2. Simple properties of the generalized Riemann integral in finite dimensional Euclidean space. 4. Integration over a fixed elementary set. 5. Integration and variation over more than one elementary set. 6. The integrability of functions of brick-point functions. 7. The variation set -- ch. 3. Limit theorems for sequences of functions. 8. Monotone convergence. 9. Bounded Riemann sums and the majorized (dominated) convergence test. 10. Controlled convergence. 11. Necessary and sufficient conditions. 12. Mean convergence and L[symbol] spaces -- ch. 4. Limit theorems for more general convergence, with continuity. 13. Basic theorems. 14. Fatou's lemma and the avoidance of nonmeasurable functions -- ch. 5. Differentiation, measurability, and inner variation. 15. Differentiation of integrals. 16. Limits of step functions -- ch. 6. Cartesian products and the Fubini and Tonelli theorems. 17. Fubini-type theorems. 18. Tonelli-type theorems and the necessary and sufficient condition for reversal of order of double integrals -- ch. 7. applications. 19. Ordinary differential equations. 20. Statistics and probability theory -- ch. 8. History and further discussion. 21. Other integrals. 22. Notes on the previous sections
Theories Of Integration
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Author : Douglas S. Kurtz
language : en
Publisher: World Scientific
Release Date : 2004
Theories Of Integration written by Douglas S. Kurtz and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.
This book presents a historical development of the integration theories of Riemann, Lebesgue, Henstock-Kurzweil, and McShane, showing how new theories of integration were developed to solve problems that earlier theories could not handle. It develops the basic properties of each integral in detail and provides comparisons of the different integrals. The chapters covering each integral are essentially independent and can be used separately in teaching a portion of an introductory course on real analysis. There is a sufficient supply of exercises to make the book useful as a textbook.
A Radical Approach To Lebesgue S Theory Of Integration
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Author : David M. Bressoud
language : en
Publisher: Cambridge University Press
Release Date : 2008-01-21
A Radical Approach To Lebesgue S Theory Of Integration written by David M. Bressoud and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-01-21 with Mathematics categories.
Meant for advanced undergraduate and graduate students in mathematics, this introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author tells the story of the mathematicians who wrestled with the difficulties inherent in the Riemann integral, leading to the work of Jordan, Borel, and Lebesgue.