# A Modern Theory of Integration (Graduate Studies in Mathematics)

## Robert Gardner Bartle

Valutazione media 3,92
( su 12 valutazioni fornite da Goodreads )

A comprehensive, beautifully written exposition of the Henstock-Kurzweil (gauge, Riemann complete) integral ... 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.

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Review:

"The book presents its subject in a pleasing, didactically surprising, and well worth reading exposition. The proofs are, as a rule, easily understandable and the significance of the theorems that are worked through is illustrated by means of numerous examples. It can be recommended as a self-study book to every student with a basic foundation in analysis. It is also very suitable as a supplementary text for a course on integration on $\mathbf{R}$." ---- Translated fromJahresbericht der Deutschen Mathematiker-Vereinigung

"A comprehensive, beautifully written exposition of the Henstock-Kurzweil (gauge, Riemann complete) integral ... There is an abundant supply of exercises which serve to make this book an excellent choice for a text for a course which would contain an elementary introduction to modern integration theory." ---- Zentralblatt MATH

Le informazioni nella sezione "Su questo libro" possono far riferimento a edizioni diverse di questo titolo.

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## 1.A MODERN THEORY OF INTEGRATION

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## 6.MODERN THEORY OF INTEGRATION, A

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Descrizione libro 2001. Soft cover. Condizione libro: New. This book is BRAND NEW Soft cover International edition with black and white printing. ISBN number & cover page may be different but contents identical to the US edition word by word. Book is in English language. Codice libro della libreria UN-STOCK-UPIN-1488

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## 7.A Modern Theory of Integration

Editore: Providence, Rhode Island, U.S.A.: Amer Mathematical Society (2001)
ISBN 10: 0821808451 ISBN 13: 9780821808450
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Descrizione libro Providence, Rhode Island, U.S.A.: Amer Mathematical Society, 2001. Soft cover. Condizione libro: New. This book is BRAND NEW Soft cover International edition with black and white printing. ISBN number and cover page may be different but contents identical to the US edition word by word. Book is in English language. Codice libro della libreria up1139

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## 8.A Modern Theory of Integration (Hardback)

Editore: American Mathematical Society, United States (2001)
ISBN 10: 0821808451 ISBN 13: 9780821808450
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Descrizione libro American Mathematical Society, United States, 2001. Hardback. Condizione libro: New. Language: English . Brand New Book. 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. Codice libro della libreria AAN9780821808450

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## 9.A Modern Theory of Integration (Hardback)

Editore: American Mathematical Society, United States (2001)
ISBN 10: 0821808451 ISBN 13: 9780821808450
Nuovi Rilegato Quantità: 1
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Descrizione libro American Mathematical Society, United States, 2001. Hardback. Condizione libro: New. Language: English . Brand New Book. 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. Codice libro della libreria AAN9780821808450

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## 10.A Modern Theory of Integration

Editore: American Mathematical Society (2001)
ISBN 10: 0821808451 ISBN 13: 9780821808450
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Descrizione libro American Mathematical Society, 2001. HRD. Condizione libro: New. New Book. Shipped from UK in 4 to 14 days. Established seller since 2000. Codice libro della libreria CE-9780821808450

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