Integration Theory - A Second Course
Integration Theory - A Second Course

Author: Martin Vaeth

Publisher: World Scientific Publishing Company

Published: 2002-08-15

Total Pages: 287

ISBN-13: 9813106034

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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.

Essentials of Integration Theory for Analysis
Essentials of Integration Theory for Analysis

Author: Daniel W. Stroock

Publisher: Springer Nature

Published: 2020-11-24

Total Pages: 296

ISBN-13: 303058478X

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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.

A Course on Integration Theory
A Course on Integration Theory

Author: Nicolas Lerner

Publisher: Springer

Published: 2014-07-09

Total Pages: 504

ISBN-13: 3034806949

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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.​

Measures, Integrals and Martingales
Measures, Integrals and Martingales

Author: René L. Schilling

Publisher: Cambridge University Press

Published: 2005-11-10

Total Pages: 404

ISBN-13: 9780521850155

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This book, first published in 2005, introduces measure and integration theory as it is needed in many parts of analysis and probability.

A Radical Approach to Lebesgue's Theory of Integration
A Radical Approach to Lebesgue's Theory of Integration

Author: David M. Bressoud

Publisher: Cambridge University Press

Published: 2008-01-21

Total Pages: 15

ISBN-13: 0521884748

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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.

Social Integration in the Second Half of Life
Social Integration in the Second Half of Life

Author: Karl Pillemer

Publisher: JHU Press

Published: 2000-11-24

Total Pages: 1014

ISBN-13: 9780801864544

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Social scientists use the term social integration to refer to individuals' connections with others in their environments. The concept and its consequences have been the subject of considerable study. Many researchers have asserted that meaningful and enduring ties to other persons serve as a buffer against stress, and thereby promote physical and mental health. The results are especially pronounced for older persons. Social Integration in the Second Half of Life presents integrative reviews of theory and research on this topic. The editors and contributors, all currently or previously affiliated with the Cornell Gerontology Research Institute, also present new empirical findings of research done at their center. The first section of the book discusses basic theory and principles of social integration in later life and its implications for health. The second, largest section examines specific issues: retirement, driving, family support, housing, neighbors. The third section addresses interventions to promote social integration: transportation, volunteering, and peer support for dementia caregivers. Throughout, the authors focus on the diverging influences of social integration and its converse, social isolation, in later life.

Geometric Integration Theory
Geometric Integration Theory

Author: Steven G. Krantz

Publisher: Springer Science & Business Media

Published: 2008-12-15

Total Pages: 344

ISBN-13: 0817646795

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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.

Theories of Integration
Theories of Integration

Author: Douglas S. Kurtz

Publisher: World Scientific

Published: 2004

Total Pages: 286

ISBN-13: 9789812388438

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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.

Measure, Integral and Probability
Measure, Integral and Probability

Author: Marek Capinski

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 229

ISBN-13: 1447136314

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This very well written and accessible book emphasizes the reasons for studying measure theory, which is the foundation of much of probability. By focusing on measure, many illustrative examples and applications, including a thorough discussion of standard probability distributions and densities, are opened. The book also includes many problems and their fully worked solutions.