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CONTENTS

PREFACE...................... III
ERRATA.......................... VII

CHAPTER I. The integral in an abstract space
§ 1. Introduction.................................. 1
§ 2. Terminology and notation...................... 4
§ 3. Abstract space X.............................. 6
§ 4. Additive classes of sets...................... 7
§ 5. Additive functions of a set................... 8
§ 6. The variations of an additive function........ 10
§ 7. Measurable functions.......................... 12
§ 8. Elementary operations on measurable functions... 14
§ 9. Measure....................................... 16
§ 10. Integral..................................... 19
§ 11. Fundamental properties of the integral....... 21
§ 12. Integration of sequences of functions........ 26
§ 13. Absolutely continuous additive functions of a set..... 30
§ 14. The Lebesgue decomposition of an additive function.... 32
§ 15. Change of measure.................................... 36

CHAPTER II. Carathéodory measure
§ 1. Preliminary remarks.................................. 39
§ 2. Metrical space....................................... 39
§ 3. Continuous and semi-continuous functions............. 42
§ 4. Carathéodory measure.................................. 43
§ 5. The operation (A)..................................... 47
§ 6. Regular sets.......................................... 50
§ 7. Borel sets............................................ 51
§ 8. Length of a set....................................... 53
§ 9. Complete space........................................ 54

CHAPTER III. Functions of bounded variation and the Lebesgue-Stieltjes integral
§ 1. Euclidean spaces....................................... 56
§ 2. Intervals and figures.................................. 57
§ 3. Functions of an interval............................... 59
§ 4. Functions of an interval that are additive and of bounded variation.... 61
§ 5. Lebesgue-Stieltjes integral. Lebesgue integral and measure.......... 64
§ 6. Measure defined by a non-negative additive function of an interval..... 67
§ 7. Theorems of Lusin and Vitali-Carathéodory.............................. 72
§ 8. Theorem of Fubini...................................................... 76
§ 9. Fubini's theorem in abstract spaces.................................... 82
§ 10. Geometrical definition of the Lebesgue-Stieltjes integral............. 88
§ 11. Translations of sets.................................................. 90
§ 12. Absolutely continuous functions of an interval....................... 93
§ 13. Functions of a real variable.......................................... 96
§ 14. Integration by parts.................................................. 102

CHAPTER IV. Derivation of additive functions of a set and of an interval
§ 1. Introduction.......................................... 105
§ 2. Derivates of functions of a set and of an interval.......................................... 106
§ 3. Vitali's Covering Theorem.......................................... 109
§ 4. Theorems on measurability of derivates.......................................... 112
§ 5. Lebesgue's Theorem.......................................... 114
§ 6. Derivation of the indefinite integral.......................................... 117
§ 7. The Lebesgue decomposition.......................................... 118
§ 8. Rectifiable curves.......................................... 121
§ 9. De la Vallée Poussin's theorem.......................................... 125
§ 10. Points of density for a set.......................................... 128
§ 11. Ward's theorems on derivation of additive functions of an interval.......................................... 133
§ 12. A theorem of Hardy-Littlewood.......................................... 142
§ 13. Strong derivation of the indefinite integral.......................................... 147
§ 14. Symmetrical derivates.......................................... 149
§ 15. Derivation in abstract spaces.......................................... 152
§ 16. Torus space.......................................... 157

CHAPTER V. Area of a surface z=F(x,y)
§ 1. Preliminary remarks.......................................... 163
§ 2. Area of a surface.......................................... 164
§ 3. The Burkill integral.......................................... 165
§ 4. Bounded variation and absolute continuity for functions of two variables.......................................... 169
§ 5. The expressions of de Geöcze.......................................... 171
§ 6. Integrals of the expressions of de Geöcze.......................................... 174
7. Radò's Theorem.......................................... 177
§ 8. Tonelli's Theorem.......................................... 181

CHAPTER VI. Major and minor functions
§ 1. Introduction.......................................... 186
§ 2. Derivation with respect to normal sequences of nets.......................................... 188
§ 3. Major and minor functions.......................................... 190
§ 4. Derivation with respect to binary sequences of nets.......................................... 191
§ 5. Applications to functions of a complex variable.......................................... 195
§ 6. The Perron integral.......................................... 201
§ 7. Derivates of functions of a real variable.......................................... 203
§ 8. The Perron-Stieltjes integral.......................................... 207

CHAPTER VII. Functions of generalized bounded variation
§ 1. Introduction.......................................... 213
§ 2. A theorem of Lusin.......................................... 215
§ 3. Approximate limits and derivatives.......................................... 218
§ 4. Functions VB and VBG.......................................... 221
§ 5. Functions AC and ACG.......................................... 223
§ 6. Lusin's condition (N).......................................... 224
§ 7. Functions VB* and VBG*.......................................... 228
§ 8. Functions AC* and ACG*.......................................... 231
§ 9. Definitions of Denjoy-Lusin.......................................... 233
§ 10. Criteria for the classes of functions VBG*, ACG*. VBG and ACG....... 234

CHAPTER VIII. Denjoy integrals
§ 1. Descriptive definition of the Denjoy integrals..................... 241
§ 2. Integration by parts.......................................... 244
§ 3. Theorem of Hake-Alexandroff-Looman.......................................... 247
§ 4. General notion of integral.......................................... 254
§ 5. Constructive definition of the Denjoy integrals.......................................... 256

CHAPTER IX. Derivates of functions of one or two real variables
§ 1. Some elementary theorems.......................................... 260
§ 2. Contingent of a set.......................................... 262
§ 3. Fundamental theorems on the contingents of plane sets.......................................... 264
§ 4. Denjoy's theorems.......................................... 269
§ 5. Relative derivates.......................................... 272
§ 6. The Banach conditions (T1) and (T2).......................................... 277
§ 7. Three theorems of Banach.......................................... 282
§ 8. Superpositions of absolutely continuous functions.......................................... 286
§ 9. The condition (D).......................................... 290
§ 10. A theorem of Denjoy-Khintchine on approximate derivates.......................................... 295
§ 11. Approximate partial derivates of functions of two variables.......................................... 297
§ 12. Total and approximate differentials.......................................... 300
§ 13. Fundamental theorems on the contingent of a set in space.......................................... 304
§ 14. Extreme differentials.......................................... 309

NOTE I by S. Banach. On Haar's measure.......................................... 314
NOTE II by S.Banach. The Lebesgue integral in abstract spaces.......................................... 320
BIBLIOGRAPHY GENERAL INDEX.......................................... 341
NOTATIONS.......................................... 344