Hardcover ISBN:  9780883853597 
Product Code:  DOL/51 
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eBook ISBN:  9781614442172 
Product Code:  DOL/51.E 
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MAA Member Price:  $45.00 
AMS Member Price:  $45.00 
Hardcover ISBN:  9780883853597 
eBook: ISBN:  9781614442172 
Product Code:  DOL/51.B 
List Price:  $125.00 $95.00 
MAA Member Price:  $93.75 $71.25 
AMS Member Price:  $93.75 $71.25 
Hardcover ISBN:  9780883853597 
Product Code:  DOL/51 
List Price:  $65.00 
MAA Member Price:  $48.75 
AMS Member Price:  $48.75 
eBook ISBN:  9781614442172 
Product Code:  DOL/51.E 
List Price:  $60.00 
MAA Member Price:  $45.00 
AMS Member Price:  $45.00 
Hardcover ISBN:  9780883853597 
eBook ISBN:  9781614442172 
Product Code:  DOL/51.B 
List Price:  $125.00 $95.00 
MAA Member Price:  $93.75 $71.25 
AMS Member Price:  $93.75 $71.25 

Book DetailsDolciani Mathematical ExpositionsVolume: 51; 2015; 325 pp
Varieties of Integration explores the critical contributions by Riemann, Darboux, Lebesgue, Henstock, Kurzweil, and Stieltjes to the theory of integration and provides a glimpse of more recent variations of the integral such as those involving operatorvalued measures. By the first year of graduate school, a young mathematician will have encountered at least three separate definitions of the integral. The associated integrals are typically studied in isolation with little attention paid to the relationships between them or to the historical issues that motivated their definitions.
This book redresses this situation by introducing the Riemann, Darboux, Lebesgue, and gauge integrals in a single volume using a common set of examples. This approach allows the reader to see how the definitions influence proof techniques and computational strategies. Then the properties of the integrals are compared in three major areas: the class of integrable functions, the convergence properties of the integral, and the best form of the Fundamental Theorems of Calculus.

Table of Contents

Chapters

Chapter 1. Historical Introduction

Chapter 2. The Riemann Integral

Chapter 3. The Darboux integral

Chapter 4. A Functional zoo

Chapter 5. Another Approach: Measure Theory

Chapter 6. The Lebesgue Integral

Chapter 7. The Gauge Integral

Chapter 8. Stieltjestype Integrals and Extensions

Chapter 9. A Look Back

Chapter 10. Afterword: $L_2$ Spaces and Fourier Series


Additional Material

Reviews

... This is a handsomely produced book (like most MAA publications), and it appears to be of just the right length and at the right level for the intended audience.
CMS Notices


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Varieties of Integration explores the critical contributions by Riemann, Darboux, Lebesgue, Henstock, Kurzweil, and Stieltjes to the theory of integration and provides a glimpse of more recent variations of the integral such as those involving operatorvalued measures. By the first year of graduate school, a young mathematician will have encountered at least three separate definitions of the integral. The associated integrals are typically studied in isolation with little attention paid to the relationships between them or to the historical issues that motivated their definitions.
This book redresses this situation by introducing the Riemann, Darboux, Lebesgue, and gauge integrals in a single volume using a common set of examples. This approach allows the reader to see how the definitions influence proof techniques and computational strategies. Then the properties of the integrals are compared in three major areas: the class of integrable functions, the convergence properties of the integral, and the best form of the Fundamental Theorems of Calculus.

Chapters

Chapter 1. Historical Introduction

Chapter 2. The Riemann Integral

Chapter 3. The Darboux integral

Chapter 4. A Functional zoo

Chapter 5. Another Approach: Measure Theory

Chapter 6. The Lebesgue Integral

Chapter 7. The Gauge Integral

Chapter 8. Stieltjestype Integrals and Extensions

Chapter 9. A Look Back

Chapter 10. Afterword: $L_2$ Spaces and Fourier Series

... This is a handsomely produced book (like most MAA publications), and it appears to be of just the right length and at the right level for the intended audience.
CMS Notices