Pesin / Birnbaum / Lukacs Classical and Modern Integration Theories

1. Auflage 2014
ISBN: 978-1-4832-6869-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 218 Seiten, Web PDF

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.

Autoren/Hrsg.

Pesin, Ivan N.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Fornt Cover
;1
2;Classical and Modern Integration Theories;4
3;Copyright Page;5
4;Dedication;6
5;Table of Contents ;8
6;FOREWORD;12
7;PREFACE;14
8;....TI.. AND TERMINOLOGY;16
9;PART I: FROM CAUCHY TO LEBESGUE;22
9.1;CHAPTER 1. FROM CAUCHY TO RIEMANN;24
9.1.1;1.1 CAUCHY'S DEFINITION OF AN INTEGRAL;24
9.1.2;1.2 RIEMANN'S DEFINITION OF THE INTEGRAL (R-INTEGRAL);27
9.1.3;1.3 UPPER AND LOWER DARBOUX INTEGRALS;30
9.2;CHAPTER 2. DEVELOPMENT OF INTEGRATION IDEAS IN THE SECOND HALF OF THE 19TH CENTURY;31
9.2.1;2.1 THE IMPROPER DIRICHLET INTEGRAL (DI-INTEGRAL);32
9.2.2;2.2 GENERALIZATION;33
9.2.3;2.3 FURTHER GENERALIZATIONS: HÖLDER'S INTEGRAL;34
9.2.4;2.4 CONTINUATION;35
9.2.5;2.5 SETS OF ZERO EXTENT;36
9.2.6;2.6 HARNACK INTEGRALS (H-INTEGRALS);39
9.2.7;2.7 DE LA VALLÉE-POUSSIN'S INTEGRAL;42
9.2.8;2.8 RELATIONSHIP BETWEEN DI- AND H-INTEGRALS;44
9.2.9;2.9 RELATIONSHIPS BETWEEN H- AND (V.P)-INTEGRALS;47
9.2.10;2.10 CONDITIONALLY CONVERGENT (V-P)-INTEGRALS;49
9.2.11;2.11 MEASURE OF SETS—PEANO-JORDAN MEASURE;50
9.2.12;2.12 PROPERTIES OF THE PEANO-JORDAN MEASURE;54
9.2.13;2.13 RIEMANN INTEGRAL-GEOMETRICAL DEFINITION;59
9.2.14;2.14 PIERPONT'S DEFINITION;60
9.2.15;2.15 INDEFIMTE INTEGRALS'' AND PRIMITIVE FUNCTIONS;61
10;PART II: THE ORIGIN OF LEBESGUE-YOUNG INTEGRATION THEORY;64
10.1;CHAPTER 3. THE BOREL MEASURE;66
10.2;CHAPTER 4. LEBESGUE'S MEASURE AND INTEGRATION;69
10.2.1;4.1 THE PROBLEM OF INTEGRATION;69
10.2.2;4.2 THE MEASUREPROBLEM;75
10.2.3;4.3 MEASURABLE FUNCTIONS;80
10.2.4;4.4 AN ANALYTICAL DEFINITION OF THE INTEGRAL;81
10.2.5;4.5 INTEGRABLE (SUMMABLE) FUNCTIONS;82
10.2.6;4.6 A GEOMETRICAL DEFINITION OF THE INTEGRAL;85
10.2.7;4.7 LEBESGUE'S INTEGRAL AND THE PROBLEM OF THE PRIMITIVE;88
10.2.8;4.8 CONCLUDING REMARKS;94
10.3;CHAPTER 5. YOUNG'S INTEGRAL;98
10.3.1;5.1 YOUNG'S INTEGRAL;98
10.3.2;5.2 YOUNG'S MEASURE THEORY;103
10.3.3;5.3 THE INTERRELATION BETWEEN LEBESGUE'S AND YOUNG'S CONTRIBUTIONS;105
10.4;CHAPTER 6. OTHER DEFINITIONS RELATED TO THE DEFINITION OF LEBESGUE'S INTEGRAL;107
10.4.1;6.1 YOUNG'S FIRST DEFINITION;108
10.4.2;6.2 BOREL'S INTEGRAL;108
10.4.3;6.3 CONTINUATION;111
10.4.4;6.4 ADDITIONAL REMARKS ON BOREL'S DEFINITIONS;114
10.4.5;6.5 RIESZ' DEFINITION;117
10.4.6;6.6 YOUNG'S SECOND DEFINITION;119
10.4.7;6.7 PIERPONT'S DEFINITION;122
10.4.8;6.8 LEBESGUE'S INTEGRAL AS LIMIT OF RIEMANN SUMS;124
10.5;Chapter 7. STIELTJES' INTEGRAL;127
10.5.1;7.1 HISTORICAL SURVEY;127
10.5.2;7.2 STIELTJES' DEFINITION;129
10.5.3;7.3 RIEMANN-STIELTJES INTEGRAL—SPECIAL FEATURES;132
10.5.4;7.4 LINEAR FUNCTIONALS—YOUNG'S DEFINITION;135
10.5.5;7.5 SET FUNCTIONS;138
10.5.6;7.6 RADON'S INTEGRAL;142
10.5.7;7.7 INTEGRALS IN ABSTRACT SPACES;147
10.5.8;7.8 CARATHÉODORY'S MEASURE;149
11;PART III: INTEGRATION IN THE SECOND DECADE OF THE 20TH CENTURY;152
11.1;CHAPTER 8. THE PROBLEM OF THE PRIMITIVE-THE DENJOY-KHINCHIN INTEGRAL;154
11.1.1;8.1 PRELIMINARY RESULTS;155
11.1.2;8.2 DENJOY'S TOTALIZATION;157
11.1.3;8.3 A DESCRIPTIVE DEFTNITION OF DENJOY INTEGRALS. KHINCHIN'S INTEGRAL;169
11.1.4;8.4 A DESCRIPTIVE DEFINITION OF THE RESTRICTED DENJOY INTEGRAL;174
11.1.5;8.5 KHINCHIN'S INVESTIGATIONS;175
11.1.6;8.6 INTERRELATIONS BETWEEN DENJOY'S INTEGRAL AND OTHER INTEGRALS;178
11.2;CHAPTER 9. PERRON'S INTEGRAL;181
11.2.1;9.1 MAJOR AND MINOR FUNCTIONS;181
11.2.2;9.2 PERRON'S INTEGRAL;184
11.2.3;9.3 REFINEMENTS;188
11.3;CHAPTER 10. DANIELL'S INTEGRAL;193
11.3.1;10.1 DANIELL'S DEFINITION;193
11.3.2;10.2 THE GENERAL CASE;196
12;CONCLUSION;199
13;REFERENCES;204
14;AUTHOR INDEX;212
15;SUBJECT INDEX;214