Advanced Integration Theory

Author : Corneliu Constantinescu
Publisher : Springer Science & Business Media
ISBN 13 : 9400708521
Total Pages : 877 pages
Book Rating : 4.25/5 ( download)

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Download or read book Advanced Integration Theory written by Corneliu Constantinescu and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 877 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Corneliu Constantinescu
Publisher : Springer
ISBN 13 : 9789400708532
Total Pages : 876 pages
Book Rating : 4.3X/5 ( download)

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Download or read book Advanced Integration Theory written by Corneliu Constantinescu and published by Springer. This book was released on 2014-09-01 with total page 876 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Robert G. Bartle
Publisher : American Mathematical Soc.
ISBN 13 : 9780821883853
Total Pages : 480 pages
Book Rating : 4.52/5 ( download)

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Download or read book A Modern Theory of Integration written by Robert G. Bartle and published by American Mathematical Soc.. This book was released on 2001-03-21 with total page 480 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Nicolas Lerner
Publisher : Springer
ISBN 13 : 3034806949
Total Pages : 504 pages
Book Rating : 4.47/5 ( download)

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Download or read book A Course on Integration Theory written by Nicolas Lerner and published by Springer. This book was released on 2014-07-09 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. The Radon-Nikodym theorem, notions of harmonic analysis, classical inequalities and interpolation theorems, including Marcinkiewicz's theorem, the definition of Lebesgue points and Lebesgue differentiation theorem are further topics included. A detailed appendix provides the reader with various elements of elementary mathematics, such as a discussion around the calculation of antiderivatives or the Gamma function. The appendix also provides more advanced material such as some basic properties of cardinals and ordinals which are useful in the study of measurability.​

Author : Hassler Whitney
Publisher : Courier Corporation
ISBN 13 : 048615470X
Total Pages : 402 pages
Book Rating : 4.01/5 ( download)

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Download or read book Geometric Integration Theory written by Hassler Whitney and published by Courier Corporation. This book was released on 2012-01-27 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geared toward upper-level undergraduates and graduate students, 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. 1957 edition.

Author : K. Weber
Publisher : Routledge
ISBN 13 : 135143778X
Total Pages : 307 pages
Book Rating : 4.83/5 ( download)

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Download or read book Integration Theory written by K. Weber and published by Routledge. This book was released on 2020-04-28 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introductory text acts as a singular resource for undergraduates learning the fundamental principles and applications of integration theory.Chapters discuss: function spaces and functionals, extension of Daniell spaces, measures of Hausdorff spaces, spaces of measures, elements of the theory of real functions on R.

Author : Eric M. Vestrup
Publisher : John Wiley & Sons
ISBN 13 : 0470317957
Total Pages : 622 pages
Book Rating : 4.52/5 ( download)

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Download or read book The Theory of Measures and Integration written by Eric M. Vestrup and published by John Wiley & Sons. This book was released on 2009-09-25 with total page 622 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible, clearly organized survey of the basic topics of measure theory for students and researchers in mathematics, statistics, and physics In order to fully understand and appreciate advanced probability, analysis, and advanced mathematical statistics, a rudimentary knowledge of measure theory and like subjects must first be obtained. The Theory of Measures and Integration illuminates the fundamental ideas of the subject-fascinating in their own right-for both students and researchers, providing a useful theoretical background as well as a solid foundation for further inquiry. Eric Vestrup's patient and measured text presents the major results of classical measure and integration theory in a clear and rigorous fashion. Besides offering the mainstream fare, the author also offers detailed discussions of extensions, the structure of Borel and Lebesgue sets, set-theoretic considerations, the Riesz representation theorem, and the Hardy-Littlewood theorem, among other topics, employing a clear presentation style that is both evenly paced and user-friendly. Chapters include: * Measurable Functions * The Lp Spaces * The Radon-Nikodym Theorem * Products of Two Measure Spaces * Arbitrary Products of Measure Spaces Sections conclude with exercises that range in difficulty between easy "finger exercises"and substantial and independent points of interest. These more difficult exercises are accompanied by detailed hints and outlines. They demonstrate optional side paths in the subject as well as alternative ways of presenting the mainstream topics. In writing his proofs and notation, Vestrup targets the person who wants all of the details shown up front. Ideal for graduate students in mathematics, statistics, and physics, as well as strong undergraduates in these disciplines and practicing researchers, The Theory of Measures and Integration proves both an able primary text for a real analysis sequence with a focus on measure theory and a helpful background text for advanced courses in probability and statistics.

Author : Daniel W. Stroock
Publisher : Springer Science & Business Media
ISBN 13 : 1475723008
Total Pages : 193 pages
Book Rating : 4.07/5 ( download)

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Download or read book A Concise Introduction to the Theory of Integration written by Daniel W. Stroock and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: This little book is the outgrowth of a one semester course which I have taught for each of the past four years at M. 1. T. Although this class used to be one of the standard courses taken by essentially every first year gradu ate student of mathematics, in recent years (at least in those when I was the instructor), the clientele has shifted from first year graduate students of mathematics to more advanced graduate students in other disciplines. In fact, the majority of my students have been from departments of engi neering (especially electrical engineering) and most of the rest have been economists. Whether this state of affairs is a reflection on my teaching, the increased importance of mathematical analysis in other disciplines, the superior undergraduate preparation of students coming to M. 1. T in mathematics, or simply the lack of enthusiasm that these students have for analysis, I have preferred not to examine too closely. On the other hand, the situation did force me to do a certain amount of thinking about what constitutes an appropriate course for a group of non-mathematicians who are courageous (foolish?) enough to sign up for an introduction to in tegration theory offered by the department of mathematics. In particular, I had to figure out what to do about that vast body of material which, in standard mathematics offerings, is "assumed to have been covered in your advanced calculus course".

Author : Heinz König
Publisher : Springer Science & Business Media
ISBN 13 : 3540618589
Total Pages : 277 pages
Book Rating : 4.84/5 ( download)

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Download or read book Measure and Integration written by Heinz König and published by Springer Science & Business Media. This book was released on 1997 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Martin Vaeth
Publisher : World Scientific Publishing Company
ISBN 13 : 9813106034
Total Pages : 288 pages
Book Rating : 4.31/5 ( download)

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Download or read book Integration Theory - A Second Course written by Martin Vaeth and published by World Scientific Publishing Company. This book was released on 2002-08-15 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a general approach to integration theory, as well as some advanced topics. It includes some new results, but is also a self-contained introduction suitable for a graduate student doing self-study or for an advanced course on integration theory.The book is divided into two parts. In the first part, integration theory is developed from the start in a general setting and immediately for vector-valued functions. This material can hardly be found in other textbooks. The second part covers various topics related to integration theory, such as spaces of measurable functions, convolutions, famous paradoxes, and extensions of formulae from elementary calculus to the setting of the Lebesgue integral.