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Author | : David M. Bressoud |
Publisher | : Cambridge University Press |
Total Pages | : 344 |
Release | : 2008-01-14 |
Genre | : Mathematics |
ISBN | : 9780521711838 |
Download A Radical Approach to Lebesgue's Theory of Integration Book in PDF, ePub and Kindle
This lively introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author stresses the original purpose of the definitions and theorems, highlighting the difficulties mathematicians encountered as these ideas were refined. The story begins with Riemann's definition of the integral, and then follows the efforts of those who wrestled with the difficulties inherent in it, until Lebesgue finally broke with Riemann's definition. With his new way of understanding integration, Lebesgue opened the door to fresh and productive approaches to the previously intractable problems of analysis.
Author | : David M. Bressoud |
Publisher | : Cambridge University Press |
Total Pages | : 15 |
Release | : 2008-01-21 |
Genre | : Mathematics |
ISBN | : 0521884748 |
Download A Radical Approach to Lebesgue's Theory of Integration Book in PDF, ePub and Kindle
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.
Author | : Thomas Hawkins |
Publisher | : Chelsea Publishing Company, Incorporated |
Total Pages | : 227 |
Release | : 2001 |
Genre | : Mathematics |
ISBN | : 9780821829639 |
Download Lebesgue's Theory of Integration Book in PDF, ePub and Kindle
In this book, Hawkins elegantly places Lebesgue's early work on integration theory within in proper historical context by relating it to the developments during the nineteenth century that motivated it and gave it significance and also to the contributions made in this field by Lebesgue's contemporaries. Hawkins was awarded the 1997 MAA Chauvenet Prize and the 2001 AMS Albert Leon Whiteman Memorial Prize for notable exposition and exceptional scholarship in the history of mathematics.
Author | : H. S. Bear |
Publisher | : Academic Press |
Total Pages | : 184 |
Release | : 2002 |
Genre | : Mathematics |
ISBN | : 9780120839711 |
Download A Primer of Lebesgue Integration Book in PDF, ePub and Kindle
The Lebesgue integral is now standard for both applications and advanced mathematics. This books starts with a review of the familiar calculus integral and then constructs the Lebesgue integral from the ground up using the same ideas. A Primer of Lebesgue Integration has been used successfully both in the classroom and for individual study. Bear presents a clear and simple introduction for those intent on further study in higher mathematics. Additionally, this book serves as a refresher providing new insight for those in the field. The author writes with an engaging, commonsense style that appeals to readers at all levels.
Author | : Gail S. Nelson |
Publisher | : American Mathematical Soc. |
Total Pages | : 233 |
Release | : 2015-11-09 |
Genre | : Mathematics |
ISBN | : 1470421992 |
Download A User-Friendly Introduction to Lebesgue Measure and Integration Book in PDF, ePub and Kindle
A User-Friendly Introduction to Lebesgue Measure and Integration provides a bridge between an undergraduate course in Real Analysis and a first graduate-level course in Measure Theory and Integration. The main goal of this book is to prepare students for what they may encounter in graduate school, but will be useful for many beginning graduate students as well. The book starts with the fundamentals of measure theory that are gently approached through the very concrete example of Lebesgue measure. With this approach, Lebesgue integration becomes a natural extension of Riemann integration. Next, -spaces are defined. Then the book turns to a discussion of limits, the basic idea covered in a first analysis course. The book also discusses in detail such questions as: When does a sequence of Lebesgue integrable functions converge to a Lebesgue integrable function? What does that say about the sequence of integrals? Another core idea from a first analysis course is completeness. Are these -spaces complete? What exactly does that mean in this setting? This book concludes with a brief overview of General Measures. An appendix contains suggested projects suitable for end-of-course papers or presentations. The book is written in a very reader-friendly manner, which makes it appropriate for students of varying degrees of preparation, and the only prerequisite is an undergraduate course in Real Analysis.
Author | : William Johnston |
Publisher | : The Mathematical Association of America |
Total Pages | : 297 |
Release | : 2015-09-25 |
Genre | : Mathematics |
ISBN | : 1939512077 |
Download The Lebesgue Integral for Undergraduates Book in PDF, ePub and Kindle
In 1902, modern function theory began when Henri Lebesgue described a new "integral calculus." His "Lebesgue integral" handles more functions than the traditional integral-so many more that mathematicians can study collections (spaces) of functions. For example, it defines a distance between any two functions in a space. This book describes these ideas in an elementary accessible way. Anyone who has mastered calculus concepts of limits, derivatives, and series can enjoy the material. Unlike any other text, this book brings analysis research topics within reach of readers even just beginning to think about functions from a theoretical point of view.
Author | : David Bressoud |
Publisher | : American Mathematical Society |
Total Pages | : 339 |
Release | : 2022-02-22 |
Genre | : Mathematics |
ISBN | : 1470469049 |
Download A Radical Approach to Real Analysis Book in PDF, ePub and Kindle
In this second edition of the MAA classic, exploration continues to be an essential component. More than 60 new exercises have been added, and the chapters on Infinite Summations, Differentiability and Continuity, and Convergence of Infinite Series have been reorganized to make it easier to identify the key ideas. A Radical Approach to Real Analysis is an introduction to real analysis, rooted in and informed by the historical issues that shaped its development. It can be used as a textbook, as a resource for the instructor who prefers to teach a traditional course, or as a resource for the student who has been through a traditional course yet still does not understand what real analysis is about and why it was created. The book begins with Fourier's introduction of trigonometric series and the problems they created for the mathematicians of the early 19th century. It follows Cauchy's attempts to establish a firm foundation for calculus and considers his failures as well as his successes. It culminates with Dirichlet's proof of the validity of the Fourier series expansion and explores some of the counterintuitive results Riemann and Weierstrass were led to as a result of Dirichlet's proof.
Author | : Thomas Sonar |
Publisher | : Springer Nature |
Total Pages | : 706 |
Release | : 2020-12-27 |
Genre | : Mathematics |
ISBN | : 303058223X |
Download 3000 Years of Analysis Book in PDF, ePub and Kindle
What exactly is analysis? What are infinitely small or infinitely large quantities? What are indivisibles and infinitesimals? What are real numbers, continuity, the continuum, differentials, and integrals? You’ll find the answers to these and other questions in this unique book! It explains in detail the origins and evolution of this important branch of mathematics, which Euler dubbed the “analysis of the infinite.” A wealth of diagrams, tables, color images and figures serve to illustrate the fascinating history of analysis from Antiquity to the present. Further, the content is presented in connection with the historical and cultural events of the respective epochs, the lives of the scholars seeking knowledge, and insights into the subfields of analysis they created and shaped, as well as the applications in virtually every aspect of modern life that were made possible by analysis.
Author | : Terence Tao |
Publisher | : American Mathematical Soc. |
Total Pages | : 206 |
Release | : 2021-09-03 |
Genre | : Education |
ISBN | : 1470466406 |
Download An Introduction to Measure Theory Book in PDF, ePub and Kindle
This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Carathéodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.
Author | : René L. Schilling |
Publisher | : Cambridge University Press |
Total Pages | : 431 |
Release | : 2021-06-17 |
Genre | : Mathematics |
ISBN | : 1009020390 |
Download Counterexamples in Measure and Integration Book in PDF, ePub and Kindle
Often it is more instructive to know 'what can go wrong' and to understand 'why a result fails' than to plod through yet another piece of theory. In this text, the authors gather more than 300 counterexamples - some of them both surprising and amusing - showing the limitations, hidden traps and pitfalls of measure and integration. Many examples are put into context, explaining relevant parts of the theory, and pointing out further reading. The text starts with a self-contained, non-technical overview on the fundamentals of measure and integration. A companion to the successful undergraduate textbook Measures, Integrals and Martingales, it is accessible to advanced undergraduate students, requiring only modest prerequisites. More specialized concepts are summarized at the beginning of each chapter, allowing for self-study as well as supplementary reading for any course covering measures and integrals. For researchers, it provides ample examples and warnings as to the limitations of general measure theory. This book forms a sister volume to René Schilling's other book Measures, Integrals and Martingales (www.cambridge.org/9781316620243).