Theory of the integral

Saks, Stanisław

  • Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa-Lwów), 1937

Abstract

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CONTENTSPREFACE...................... IIIERRATA.......................... VIICHAPTER 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.................................... 36CHAPTER 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........................................ 54CHAPTER 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.................................................. 102CHAPTER 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.......................................... 157CHAPTER 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.......................................... 1747. Radò's Theorem.......................................... 177§ 8. Tonelli's Theorem.......................................... 181CHAPTER 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.......................................... 207CHAPTER 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....... 234CHAPTER 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.......................................... 256CHAPTER 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.......................................... 309NOTE I by S. Banach. On Haar's measure.......................................... 314NOTE II by S.Banach. The Lebesgue integral in abstract spaces.......................................... 320BIBLIOGRAPHY GENERAL INDEX.......................................... 341NOTATIONS.......................................... 344

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Saks, Stanisław. Theory of the integral. Warszawa-Lwów: Instytut Matematyczny Polskiej Akademi Nauk, 1937. <http://eudml.org/doc/219302>.

@book{Saks1937,
abstract = {CONTENTSPREFACE...................... IIIERRATA.......................... VIICHAPTER 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.................................... 36CHAPTER 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........................................ 54CHAPTER 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.................................................. 102CHAPTER 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.......................................... 157CHAPTER 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.......................................... 1747. Radò's Theorem.......................................... 177§ 8. Tonelli's Theorem.......................................... 181CHAPTER 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.......................................... 207CHAPTER 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....... 234CHAPTER 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.......................................... 256CHAPTER 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.......................................... 309NOTE I by S. Banach. On Haar's measure.......................................... 314NOTE II by S.Banach. The Lebesgue integral in abstract spaces.......................................... 320BIBLIOGRAPHY GENERAL INDEX.......................................... 341NOTATIONS.......................................... 344},
author = {Saks, Stanisław},
keywords = {Set theory, real functions},
language = {eng},
location = {Warszawa-Lwów},
publisher = {Instytut Matematyczny Polskiej Akademi Nauk},
title = {Theory of the integral},
url = {http://eudml.org/doc/219302},
year = {1937},
}

TY - BOOK
AU - Saks, Stanisław
TI - Theory of the integral
PY - 1937
CY - Warszawa-Lwów
PB - Instytut Matematyczny Polskiej Akademi Nauk
AB - CONTENTSPREFACE...................... IIIERRATA.......................... VIICHAPTER 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.................................... 36CHAPTER 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........................................ 54CHAPTER 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.................................................. 102CHAPTER 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.......................................... 157CHAPTER 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.......................................... 1747. Radò's Theorem.......................................... 177§ 8. Tonelli's Theorem.......................................... 181CHAPTER 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.......................................... 207CHAPTER 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....... 234CHAPTER 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.......................................... 256CHAPTER 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.......................................... 309NOTE I by S. Banach. On Haar's measure.......................................... 314NOTE II by S.Banach. The Lebesgue integral in abstract spaces.......................................... 320BIBLIOGRAPHY GENERAL INDEX.......................................... 341NOTATIONS.......................................... 344
LA - eng
KW - Set theory, real functions
UR - http://eudml.org/doc/219302
ER -

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  1. Paul-André Meyer, Théorèmes fondamentaux du calcul des probabilités
  2. Martin Zerner, Diverses conséquences d'une formule de Green
  3. Jiří Jarník, Jaroslav Kurzweil, A general form of the product integral and linear ordinary differential equations
  4. A. M. Bruckner, M. Rosenfeld, A theorem on approximate directional derivatives
  5. Jiří Štulc, O délce křivek a kontinuí
  6. Štefan Schwabik, On the relation between Young's and Kurzweil's concept of Stieltjes integral
  7. W. J. Trjitzinsky, La régularité moyenne dans la théorie métrique
  8. Serban Teodor Belinschi, A note on regularity for free convolutions
  9. Jaroslav Kurzweil, Štefan Schwabik, Ordinary differential equations the solution of which are A C G * -functions
  10. Štefan Schwabik, On a modified sum integral of Stieltjes type

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