# Classical and Modern Integration Theories

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Classical and Modern Integration Theories discusses classical integration theory, particularly that part of the theory directly associated with the problems of area. The book reviews the history and the determination of primitive functions, beginning from Cauchy to Daniell. The text describes Cauchy's definition of an integral, Riemann's definition of the R-integral, the upper and lower Darboux integrals. The book also reviews the origin of the Lebesgue-Young integration theory, and Borel's postulates that define measures of sets. W.H. Young's work provides a construction of the integral equivalent to Lebesque's construction with a different generalization of integrals leading to different approaches in solutions. Young's investigations aim at generalizing the notion of length for arbitrary sets by means of a process which is more general than Borel's postulates. The text notes that the Lebesgue measure is the unique solution of the measure problem for the class of L-measurable sets. The book also describes further modifications made into the Lebesgue definition of the integral by Riesz, Pierpont, Denjoy, Borel, and Young. These modifications bring the Lebesgue definition of the integral closer to the Riemann or Darboux definitions, as well as to have it associated with the concepts of classical analysis. The book can benefit mathematicians, students, and professors in calculus or readers interested in the history of classical mathematics.

## Table of Contents

Foreword

Preface

Notation and Terminology

I. From Cauchy To Lebesgue

1. From Cauchy to Riemann

1.1 Cauchy's Definition of an Integral

1.2 Riemann's Definition of the Integral (R-Integral)

1.3 Upper and Lower Darboux Integrals

2. Development of Integration Ideas in the Second Half of the 19th Century

2.1 The Improper Dirichlet Integral (Di-Integral)

2.2 Generalization

2.3 Further Generalizations: Hölder's Integral

2.4 Continuation

2.5 Sets of Zero Extent

2.6 Harnack Integrals (H-Integrals)

2.7 De la Vallée-Poussin's Integral

2.8 Relationship Between Di- and H-Integrals

2.9 Relationships Between H- and (V-P)-Integrals

2.10 Conditionally Convergent (V-P)-Integrals

2.11 Measure of Sets — Peano-Jordan Measure

2.12 Properties of the Peano-Jordan Measure

2.13 Riemann Integral—Geometrical Definition

2.14 Pierpont's Definition

2.15 Indefinite Integrals and Primitive Functions

II. The Origin Of Lebesgue-Young Integration Theory

3. The Borel Measure

4. Lebesgue's Measure and Integration

4.1 The Problem of Integration

4.2 The Measure Problem

4.3 Measurable Functions

4.4 Analytical Definition of the Integral

4.5 Integrable (Summable) Functions

4.6 A Geometrical Definition of the Integral

4.7 Lebesgue's Integral and the Problem of the Primitive

4.8 Concluding Remarks

5. Young's Integral

5.1 Young's Integral

5.2 Young's Measure Theory

5.3 The Interrelation Between Lebesgue's and Young's Contributions

6. Other Definitions Related to the Definition of Lebesgue's Integral

6.1 Young's First Definition

6.2 Borel's Integral

6.3 Continuation

6.4 Additional Remarks on Borel's Definitions

6.5 Riesz' Definition

6.6 Young's Second Definition

6.7 Pierpont's Definition

6.8 Lebesgue's Integral as Limit of Riemann Sums

7. Stieltjes' Integral

7.1 Historical Survey

7.2 Stieltjes' Definition

7.3 Riemann-Stieltjes Integral — Specific Features

7.4 Linear Functionals—Young's Definition

7.5 Set Functions

7.6 Radon's Integral

7.7 Integrals in Abstract Spaces

7.8 Carathéodory's Measure

III. Ιntegration In The Second Decade Of The 20th Century

8. The Problem of the Primitive—The Denjoy-Khinchin Integral

8.1 Preliminary Results

8.2 Denjoy's Totalization

8.3 A Descriptive Definition of Denjoy Integrals. Khinchin's Integral

8.4 A Descriptive Definition of the Restricted Denjoy Integral

8.5 Khinchin's Investigations

8.6 Interrelations Between Denjoy's Integral and Other Integrals

9· Perron's Integral

9.1 Major and Minor Functions

9.2 Perron's Integral

9.3 Refinements

10. Darnell's Integral

10.1 Daniell's Definition

10.2 The General Case

Conclusion

References

Author Index

Subject Index

## Product details

- No. of pages: 218
- Language: English
- Copyright: © Academic Press 1970
- Published: January 28, 1970
- Imprint: Academic Press
- eBook ISBN: 9781483268699

## About the Author

### Ivan N. Pesin

## About the Editors

### Z. W. Birnbaum

### E. Lukacs

#### Affiliations and Expertise

Bowling Green State University

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