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## Theory of the integral

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Monografie Matematyczne tom/nr w serii: 7. wydano: 1937
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PREFACE ERRATA CHAPTER I. The integral in an abstract space § 1. Introduction § 2. Terminology and notation § 3. Abstract space X § 4. Additive classes of sets § 5. Additive functions of a set § 6. The variations of an additive function § 7. Measurable functions § 8. Elementary operations on measurable functions § 9. Measure § 10. Integral § 11. Fundamental properties of the integral § 12. Integration of sequences of functions § 13. Absolutely continuous additive functions of a set § 14. The Lebesgue decomposition of an additive function § 15. Change of measure CHAPTER II. Carathéodory measure § 1. Preliminary remarks § 2. Metrical space § 3. Continuous and semi-continuous functions § 4. Carathéodory measure § 5. The operation (A) § 6. Regular sets § 7. Borel sets § 8. Length of a set § 9. Complete space CHAPTER III. Functions of bounded variation and the Lebesgue-Stieltjes integral § 1. Euclidean spaces § 2. Intervals and figures § 3. Functions of an interval § 4. Functions of an interval that are additive and of bounded variation § 5. Lebesgue-Stieltjes integral. Lebesgue integral and measure § 6. Measure defined by a non-negative additive function of an interval § 7. Theorems of Lusin and Vitali-Carathéodory § 8. Theorem of Fubini § 9. Fubini's theorem in abstract spaces § 10. Geometrical definition of the Lebesgue-Stieltjes integral § 11. Translations of sets § 12. Absolutely continuous functions of an interval § 13. Functions of a real variable § 14. Integration by parts CHAPTER IV. Derivation of additive functions of a set and of an interval § 1. Introduction § 2. Derivates of functions of a set and of an interval § 3. Vitali's Covering Theorem § 4. Theorems on measurability of derivates § 5. Lebesgue's Theorem § 6. Derivation of the indefinite integral § 7. The Lebesgue decomposition § 8. Rectifiable curves § 9. De la Vallée Poussin's theorem § 10. Points of density for a set § 11. Ward's theorems on derivation of additive functions of an interval § 12. A theorem of Hardy-Littlewood § 13. Strong derivation of the indefinite integral § 14. Symmetrical derivates § 15. Derivation in abstract spaces § 16. Torus space CHAPTER V. Area of a surface z=F(x,y) § 1. Preliminary remarks § 2. Area of a surface § 3. The Burkill integral § 4. Bounded variation and absolute continuity for functions of two variables § 5. The expressions of de Geöcze § 6. Integrals of the expressions of de Geöcze § 7. Radò's Theorem § 8. Tonelli's Theorem CHAPTER VI. Major and minor functions § 1. Introduction § 2. Derivation with respect to normal sequences of nets § 3. Major and minor functions § 4. Derivation with respect to binary sequences of nets § 5. Applications to functions of a complex variable § 6. The Perron integral § 7. Derivates of functions of a real variable § 8. The Perron-Stieltjes integral CHAPTER VII. Functions of generalized bounded variation § 1. Introduction § 2. A theorem of Lusin § 3. Approximate limits and derivatives § 4. Functions VB and VBG § 5. Functions AC and ACG § 6. Lusin's condition (N) § 7. Functions VB* and VBG* § 8. Functions AC* and ACG* § 9. Definitions of Denjoy-Lusin CHAPTER VIII. Denjoy integrals § 1. Descriptive definition of the Denjoy integrals § 2. Integration by parts § 3. Theorem of Hake-Alexandroff-Looman § 4. General notion of integral § 5. Constructive definition of the Denjoy integrals CHAPTER IX. Derivates of functions of one or two real variables § 1. Some elementary theorems § 2. Contingent of a set § 3. Fundamental theorems on the contingents of plane sets § 4. Denjoy's theorems § 5. Relative derivates § 6. The Banach conditions (T1) and (T2) § 7. Three theorems of Banach § 8. Superpositions of absolutely continuous functions § 9. The condition (D) § 10. A theorem of Denjoy-Khintchine on approximate derivates § 11. Approximate partial derivates of functions of two variables § 12. Total and approximate differentials § 13. Fundamental theorems on the contingent of a set in space § 14. Extreme differentials NOTE I by S. Banach. On Haar's measure NOTE II by S.Banach. The Lebesgue integral in abstract spaces BIBLIOGRAPHY GENERAL INDEX NOTATIONS
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Monografie Matematyczne, Tom 7
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1937
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