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. <P>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. <P>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. <P>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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"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
"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
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5 of 7 people found the following review helpful:
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Nice intro to gauge integrals,
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This review is from: A Modern Theory of Integration (Graduate Studies in Mathematics) (Hardcover)
Excellent introduction to the theory of gauge integrals. A mild acquaintance with riemann integrals is all that's required to get started with this book.
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First Sentence:
The symbol R always denotes the real number system, the properties of which we assume to be familiar to the reader. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
tagged partition, nondegenerate compact interval, semimetric space, pairwise disjoint sequence, integrable set, null function, regulated function, absolutely integrable functions, indefinite integral, limit inferior, null set, absolute integrability, constant gauge, countable collection, convergence theorem, compact intervals, further subsequence, exceptional set, singular function, measurable set, measurable functions, unbounded interval, nonoverlapping intervals, pairwise disjoint sets, negligible variation Key Phrases - Capitalized Phrases (CAPs): (learn more)
Hake's Theorem, Fundamental Theorem, Monotone Convergence Theorem, Dominated Convergence Theorem, Egorov's Theorem, Fatou's Lemma, Differentiation Theorem, Cauchy Criterion, Saks-Henstock Lemma, Vitali Covering Theorem, Characterization Theorem, First Substitution Theorem, Mean Convergence Theorem, Multiplier Theorem, Completeness Theorem, Second Mean Value Theorem, Second Substitution Theorem, Straddle Lemma, Abel's Test, Density Theorem, Increasing Sequence Theorem, Measurable Limit Theorem, Riesz Subsequence Theorem, Additivity Theorem, Approximation Theorem New!
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The symbol R always denotes the real number system, the properties of which we assume to be familiar to the reader. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
tagged partition, nondegenerate compact interval, semimetric space, pairwise disjoint sequence, integrable set, null function, regulated function, absolutely integrable functions, indefinite integral, limit inferior, null set, absolute integrability, constant gauge, countable collection, convergence theorem, compact intervals, further subsequence, exceptional set, singular function, measurable set, measurable functions, unbounded interval, nonoverlapping intervals, pairwise disjoint sets, negligible variation Key Phrases - Capitalized Phrases (CAPs): (learn more)
Hake's Theorem, Fundamental Theorem, Monotone Convergence Theorem, Dominated Convergence Theorem, Egorov's Theorem, Fatou's Lemma, Differentiation Theorem, Cauchy Criterion, Saks-Henstock Lemma, Vitali Covering Theorem, Characterization Theorem, First Substitution Theorem, Mean Convergence Theorem, Multiplier Theorem, Completeness Theorem, Second Mean Value Theorem, Second Substitution Theorem, Straddle Lemma, Abel's Test, Density Theorem, Increasing Sequence Theorem, Measurable Limit Theorem, Riesz Subsequence Theorem, Additivity Theorem, Approximation Theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 2
books:
5
books
cite this book:
- Real Analysis by Andrew M. Bruckner in Back Matter
- Real Analysis (Dover Books on Mathematics) by Edward James McShane in Back Matter
- Measure Theory and Integration (Chapman & Hall/CRC Pure and Applied Mathematics) by M.M. Rao in Back Matter
- Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao (Lecture Notes in Pure and Applied Mathematics) by Alan C. Krinik on page 117
- An Introduction to Measure and Integration (Graduate Studies in Mathematics) by Inder K. Rana in Back Matter
- Introduction to Real Analysis, 3rd Edition by Robert G. Bartle in Back Matter
- Uncertain Input Data Problems and the Worst Scenario Method, Volume 46 (North-Holland Series in Applied Mathematics and Mechanics) by Ivan Hlavacek in Back Matter
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