Integration

Book 7 in the Princeton Mathematical Series. Originally published in 1961. The **Princeton Legacy Library** uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. Preface Contents I. SOME THEOREMS ON REAL-VALUED FUNCTIONS 1. Sets and characteristic functions 2. Neighborhoods, openness, closure 3. Denumerable sets 4. Functions and limits 5. Bounds 6. Upper and lower limits 7. Semi-continuous functions 8. Functions of bounded variation 9. Absolutely continuous functions II. THE LEBESGUE INTEGRAL 10. Step-functions 11. Riemann integrals 12. U-functions and L-functions 13. Integrals of U-functions and L-functions 14. Upper and lower integrals 15. The Lebesgue integral 7 16. Consistency of Riemann and Lebesgue integrals 17. Integrals over bounded sets 18. Integrals over unbounded sets III. MEASURABLE SETS AND MEASURABLE FUNCTIONS 19. Arithmetic of measurable sets 20. Exterior measure and interior measure 21. Measurablefunctions 22. Measurable functions and summable functions 23. Equivalence of functions 24. Summable products. Inequalities IV. THE INTEGRAL AS A FUNCTION OF SETS; CONVERGENCE THEOREMS 25. Multiple integrals and iterated integrals 26. Set functions 27. The integral as a set function 28. Modes of convergence 29. Convergence theorems 30. Metric spaces; spaces Lv V. DIFFERENTIATION 31. Dini derivates 32. Derivates of monotonic functions 33. Derivatives of indefinite Lebesgue integrals 34. Derivatives of functions of bounded variation 35. Derivatives of absolutely continuous functions 36. Integration by parts 37. Mean-value theorems 38. Substitutiontheorems 39. Differentiation under the integral sign VI. CONTINUITY PROPERTIES OF MEASURABLE FUNCTIONS 40. The classes of Baire 41. Metric density and approximate continuity 42. Density of continuous functions in Lv. Riemann-Lebesgue theorem 43. Lusin's theorem 44. Non-measurable sets and non-measurable functions VII. THE LEBESGUE-STIELTJES INTEGRAL 45. The difference-function 46. Monotonic functions and functions of bounded variation 47. Integrals and measure with respect to monotonic functions 48. Examples 49. Borel sets 50. Dependence of integral on integrator 51. Integrals with respect to functions of bounded variation 52. PropertiesoftheLebesgue-Stieltjesintegral 53. Measure functions 54. Measure functions and Lebesgue-Stieltjes measure 55. Integrals with respect to a measure function 56. Measure functions defined by integrals VIII. THE PERRON INTEGRAL 57. Definition of the Perron integral 58. Elementary properties 59. Relation to the Lebesgue integral 60. Perron integral of a derivative 61. Derivative of the indefinite Perron integral 62. Summability of non-negative integrable functions 63. Convergence theorems 64. Substitution 65. Integration by parts 66. Second theorem of mean value IX. DIFFERENTIAL EQUATIONS 67. Ascoli's theorem 68. Existence and uniqueness of solutions 69. The solutions as functions of parameters X. DIFFERENTIATION OF MULTIPLE INTEGRALS 70. Vitali's theorem 71. Derivates of set functions 72. Derivatives of indefinite integrals 73. Derivatives of functions of bounded variation APPENDIX List of special symbols and abbreviations INDEX Book 7 in the Princeton Mathematical Series. Originally published in 1961. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905