Classical and Modern Integration Theories

ForewordPrefaceNotation and TerminologyI. 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 FunctionsII. 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 MeasureIII. 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 CaseConclusionReferencesAuthor IndexSubject Index