Theory of the integral

Stanisław Saks

GDML_Books, (1937), p.

Résumé

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

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