The Riemann Approach To Integration

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The Riemann Approach To Integration

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Author : Washek F. Pfeffer
language : en
Publisher: Cambridge University Press
Release Date : 1993

The Riemann Approach To Integration written by Washek F. Pfeffer 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 1993 with Mathematics categories.

A detailed exposition of generalised Riemann-Stieltjes integrals.

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.

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.

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.

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.

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

Riemann S Method Of Integration

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Author : E. W. Montroll
language : en
Publisher:
Release Date : 1952

Riemann S Method Of Integration written by E. W. Montroll and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1952 with Compressors categories.

Elementary Analysis

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Author : Kenneth A. Ross
language : en
Publisher: CUP Archive
Release Date : 2014-01-15

Elementary Analysis written by Kenneth A. Ross and has been published by CUP Archive this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with Mathematics categories.

Derivation And Integration

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Author : Washek F. Pfeffer
language : en
Publisher: Cambridge University Press
Release Date : 2001-03-05

Derivation And Integration written by Washek F. Pfeffer 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 2001-03-05 with Mathematics categories.

This 2001 book is devoted to an invariant multidimensional process of recovering a function from its derivative. It considers additive functions defined on the family of all bounded BV sets that are continuous with respect to a suitable topology. A typical example is the flux of a continuous vector field. A very general Gauss-Green theorem follows from the sufficient conditions for the derivability of the flux. Since the setting is invariant with respect to local lipeomorphisms, a standard argument extends the Gauss-Green theorem to the Stokes theorem on Lipschitz manifolds. In addition, the author proves the Stokes theorem for a class of top-dimensional normal currents - a first step towards solving a difficult open problem of derivation and integration in middle dimensions. The book contains complete and detailed proofs and will provide valuable information to research mathematicians and advanced graduate students interested in geometric integration and related areas.