| CHAPTER Ⅰ. The integral in an abst 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. Caratheodory measure. 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 CHAPTER Ⅲ. Functions of bounded variation and the Lebesgue-Stielties 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-Caratheodory 8. Theorem of Fubini 9. Fubini's theorem in abst 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 Vallee 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 abst spaces I6. 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 GeSeze 6. Integrals of the expressions of de GeScze 7. RadS's Theorem 8. Tonelli's Theorem CHAPTER Ⅵ. Major and minor tmetlons 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 Ⅵ Functions of generalized bounded variation CHAPTER Ⅷ Denjoy integrals CHAPTER Ⅸ Derivates of functions of one or two real variables NOTE Ⅰ by S.BANACH.On Haar's measure NOTE Ⅱ by S.BANACH.The Lebesgue integral in abst spaces BIBLIOGRAPHY GENERAL INDEX |
内容加载中，请稍后...
商品评论（0条）