- PREFACE [PDF]
ERRATA [PDF]
CHAPTER I. [PDF]
The integral in an abstract space.

- § 1. Introduction
§ 2. Terminology and notation
§ 3. Abstract space

- § 1. Preliminary remarks
§ 2. Metrical space
§ 3. Continuous and semi-continuous functions
§ 4. Caratheodory measure
§ 5. The operation (A)
§ 6. Regular sets
§ 7. Borel sets
§ 8. Length of a set
§ 9. Complete space

- § 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-Caratheodory
§ 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

- § 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 Vallee Poussin's theorem
§ 10. Points of density for a set
§ 11. Ward's theorems on derivation of additive functions of an
interval 133
§ 12. A theorem of Hardy-Littlewood
§ 13. Strong derivation of the indefinite integral
§ 14. Symmetrical derivates
§ 15. Derivation in abstract spaces
§ 16. Torus space

- § 1. Preliminary remarks
§ 2. Area of a surface
§ 3. The Burkill integral
§ 4. Bounded variation and absolute continuity for functions of
two variables 169
§ 5. The expressions of de Geöcze
§ 6. Integrals of the expressions of de Geöcze
§ 7. Rado's Theorem
§ 8. Tonelli's Theorem

- § 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

- § 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 ACCr*
§ 9. Definitions of Denjoy-Luain
§ 10. Criteria for the classes of functions VBG*, ACG*, VBG and
ACG

- § 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

- § 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 (T

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Emilia Jakimowicz i Adam MiranowiczFile translated from T

On 04 Jan 2012, 18:52.