A Course On Integration Theory

Author : Nicolas Lerner
Publisher : Springer
Page : 504 pages
File Size : 53,7 Mb
Release : 2014-07-09
Category : Mathematics
ISBN : 9783034806947

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A Course on Integration Theory by Nicolas Lerner Pdf

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 : Daniel W. Stroock
Publisher : Springer Nature
Page : 296 pages
File Size : 46,8 Mb
Release : 2020-11-24
Category : Mathematics
ISBN : 9783030584788

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Essentials of Integration Theory for Analysis by Daniel W. Stroock Pdf

When the first edition of this textbook published in 2011, it constituted a substantial revision of the best-selling Birkhäuser title by the same author, A Concise Introduction to the Theory of Integration. Appropriate as a primary text for a one-semester graduate course in integration theory, this GTM is also useful for independent study. A complete solutions manual is available for instructors who adopt the text for their courses. This second edition has been revised as follows: §2.2.5 and §8.3 have been substantially reworked. New topics have been added. As an application of the material about Hermite functions in §7.3.2, the author has added a brief introduction to Schwartz's theory of tempered distributions in §7.3.4. Section §7.4 is entirely new and contains applications, including the Central Limit Theorem, of Fourier analysis to measures. Related to this are subsections §8.2.5 and §8.2.6, where Lévy's Continuity Theorem and Bochner's characterization of the Fourier transforms of Borel probability on RN are proven. Subsection 8.1.2 is new and contains a proof of the Hahn Decomposition Theorem. Finally, there are several new exercises, some covering material from the original edition and others based on newly added material.

Author : Daniel W. Stroock
Publisher : Springer Science & Business Media
Page : 193 pages
File Size : 51,8 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781475723007

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A Concise Introduction to the Theory of Integration by Daniel W. Stroock Pdf

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 : K. Chandrasekharan
Publisher : Unknown
Page : 118 pages
File Size : 48,5 Mb
Release : 1996
Category : Integrals, Generalized
ISBN : 8185931097

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A Course on Integration Theory by K. Chandrasekharan Pdf

Author : Steven G. Krantz,Harold R. Parks
Publisher : Springer Science & Business Media
Page : 340 pages
File Size : 41,8 Mb
Release : 2008-12-15
Category : Mathematics
ISBN : 9780817646790

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Geometric Integration Theory by Steven G. Krantz,Harold R. Parks Pdf

This textbook introduces geometric measure theory through the notion of currents. Currents, continuous linear functionals on spaces of differential forms, are a natural language in which to formulate types of extremal problems arising in geometry, and can be used to study generalized versions of the Plateau problem and related questions in geometric analysis. Motivating key ideas with examples and figures, this book is a comprehensive introduction ideal for both self-study and for use in the classroom. The exposition demands minimal background, is self-contained and accessible, and thus is ideal for both graduate students and researchers.

Author : Heinz Bauer
Publisher : Walter de Gruyter
Page : 249 pages
File Size : 49,5 Mb
Release : 2011-04-20
Category : Mathematics
ISBN : 9783110866209

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Measure and Integration Theory by Heinz Bauer Pdf

This book gives a straightforward introduction to the field as it is nowadays required in many branches of analysis and especially in probability theory. The first three chapters (Measure Theory, Integration Theory, Product Measures) basically follow the clear and approved exposition given in the author's earlier book on "Probability Theory and Measure Theory". Special emphasis is laid on a complete discussion of the transformation of measures and integration with respect to the product measure, convergence theorems, parameter depending integrals, as well as the Radon-Nikodym theorem. The final chapter, essentially new and written in a clear and concise style, deals with the theory of Radon measures on Polish or locally compact spaces. With the main results being Luzin's theorem, the Riesz representation theorem, the Portmanteau theorem, and a characterization of locally compact spaces which are Polish, this chapter is a true invitation to study topological measure theory. The text addresses graduate students, who wish to learn the fundamentals in measure and integration theory as needed in modern analysis and probability theory. It will also be an important source for anyone teaching such a course.

Author : Luigi Ambrosio,Giuseppe Da Prato,Andrea Mennucci
Publisher : Springer Science & Business Media
Page : 193 pages
File Size : 51,7 Mb
Release : 2012-02-21
Category : Mathematics
ISBN : 9788876423864

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Introduction to Measure Theory and Integration by Luigi Ambrosio,Giuseppe Da Prato,Andrea Mennucci Pdf

This textbook collects the notes for an introductory course in measure theory and integration. The course was taught by the authors to undergraduate students of the Scuola Normale Superiore, in the years 2000-2011. The goal of the course was to present, in a quick but rigorous way, the modern point of view on measure theory and integration, putting Lebesgue's Euclidean space theory into a more general context and presenting the basic applications to Fourier series, calculus and real analysis. The text can also pave the way to more advanced courses in probability, stochastic processes or geometric measure theory. Prerequisites for the book are a basic knowledge of calculus in one and several variables, metric spaces and linear algebra. All results presented here, as well as their proofs, are classical. The authors claim some originality only in the presentation and in the choice of the exercises. Detailed solutions to the exercises are provided in the final part of the book.

Author : Heinz König
Publisher : Springer Science & Business Media
Page : 277 pages
File Size : 50,5 Mb
Release : 1997
Category : Mathematics
ISBN : 9783540618584

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Measure and Integration by Heinz König Pdf

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 : Robert G. Bartle
Publisher : American Mathematical Soc.
Page : 480 pages
File Size : 40,5 Mb
Release : 2001-03-21
Category : Electronic
ISBN : 0821883852

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A Modern Theory of Integration by Robert G. Bartle Pdf

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 : Martin Väth
Publisher : Unknown
Page : 128 pages
File Size : 42,8 Mb
Release : 2002
Category : Electronic
ISBN : 9812776826

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Integration Theory by Martin Väth Pdf

Author : K. Chandrasekharan
Publisher : Springer
Page : 125 pages
File Size : 48,9 Mb
Release : 1996-01-01
Category : Mathematics
ISBN : 9789380250885

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A Course on Integration Theory by K. Chandrasekharan Pdf

Author : Vladimir Kadets
Publisher : Springer
Page : 539 pages
File Size : 53,8 Mb
Release : 2018-07-10
Category : Mathematics
ISBN : 9783319920047

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A Course in Functional Analysis and Measure Theory by Vladimir Kadets Pdf

Written by an expert on the topic and experienced lecturer, this textbook provides an elegant, self-contained introduction to functional analysis, including several advanced topics and applications to harmonic analysis. Starting from basic topics before proceeding to more advanced material, the book covers measure and integration theory, classical Banach and Hilbert space theory, spectral theory for bounded operators, fixed point theory, Schauder bases, the Riesz-Thorin interpolation theorem for operators, as well as topics in duality and convexity theory. Aimed at advanced undergraduate and graduate students, this book is suitable for both introductory and more advanced courses in functional analysis. Including over 1500 exercises of varying difficulty and various motivational and historical remarks, the book can be used for self-study and alongside lecture courses.

Author : Douglas S. Kurtz,Jaroslav Kurzweil,Charles W. Swartz
Publisher : World Scientific
Page : 286 pages
File Size : 43,7 Mb
Release : 2004
Category : Mathematics
ISBN : 9812388435

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Theories of Integration by Douglas S. Kurtz,Jaroslav Kurzweil,Charles W. Swartz Pdf

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.

Author : Angus Ellis Taylor
Publisher : Courier Corporation
Page : 451 pages
File Size : 44,8 Mb
Release : 1985-01-01
Category : Mathematics
ISBN : 9780486649887

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General Theory of Functions and Integration by Angus Ellis Taylor Pdf

Uniting a variety of approaches to the study of integration, a well-known professor presents a single-volume "blend of the particular and the general, of the concrete and the abstract." 1966 edition.

Author : Sheldon Axler
Publisher : Springer Nature
Page : 430 pages
File Size : 55,6 Mb
Release : 2019-11-29
Category : Mathematics
ISBN : 9783030331436

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Measure, Integration & Real Analysis by Sheldon Axler Pdf

This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online. For errata and updates, visit https://measure.axler.net/