A Course of Higher Mathematics

IntroductionPrefaceChapter The Stieltjes Integral 1. Sets and their Powers 2. The Stieltjes Integral and its Basic Properties 3. Darboux Sums 4. The Stieltjes Integral of a Continuous Function 5. The Improper Stieltjes Integral 6. Jump Functions 7. Physical Interpretation 8. Functions of Bounded Variation 9. An Integrating Function of Bounded Variation 10. Existence of the Stieltjes Integral 11. Passage to the Limit in the Stieltjes Integral 12. Helly's Theorem 13. Selection Principle 14. Space of Continuous Functions 15. General Form of the Functional in Space G 16. Linear Operators in G 17. Functions of an Interval 18. The General Stieltjes Integral 19. Properties of the (General) Stieltjes Integral 20. The Existence of the General Stieltjes Integral 21. Functions of a Two-Dimensional Interval 22. Passage to Point Functions 23. The Stieltjes Integral on a Plane 24. Functions of Bounded Variation on the Plane 25. The Space of Continuous Functions of Several Variables 26. the Fourier-Stieltjes Integral 27. Inversion Formula 28. Convolution Theorem 29. The Cauchy-Stieltjes IntegralChapter II Set Functions and the Lebesgue Integral § 1. Set Functions and the Theory of Measure 30. Operations on Sets 31. Point Sets 32. Properties of Closed and Open Sets 33. Elementary Figures 34. Exterior Measure and its Properties 35. Measurable Sets 36. Measurable Sets (Continued) 37. Criteria for Measurability 38. Field of Sets 39. Independence of the Choice of Axes 40. The B Field 41. The Case of a Single Variable § 2. Measurable Functions 42. Definition of Measurable Function 43. Properties of Measurable Functions 44. the Limit of a Measurable Function 45. the G Property 46. Piecewise Constant Functions 47. Class B § 3. The Lebesgue Integral 48. The Integral of a Bounded Function 49. Properties of the Integral 50. The Integral of a Non-Negative Unbounded Function 51. Properties of the Integral 52. Functions of Any Sign 53. Complex Summable Functions 54. Passage to the Limit Under the Integral Sign 55. The Class L2 56. Convergence in the Mean 57. Hubert Function Space. Orthogonal Systems of Functions 59. The Space U 60. Lineals in 58. L2 61. Examples of Closed Systems 62. The Holder and Minkovskii Inequalities 63. Integral Over a Set of Infinite Measure 64. The Class L2 on a Set of Infinite Measure 65. An Integrating Function of Bounded Variation 66. The Reduction of Multiple Integrals 67. The Case of the Characteristic Function 68. Fubini's Theorem 69. Change of the Order of Integration 70. Continuity in the Mean 71. Mean FunctionsChapter III Set Functions. Absolute Continuity. Generalization of the Integral 72. Additive Set Functions 73. Singular Function 74. The Case of One Variable 75. Absolutely Continuous Set Functions 76. Example 77. Absolutely Continuous Functions of Several Variables 78. Supplementary Propositions 79. Supplementary Propositions (Continued) 80. Fundamental Theorem 81. Hellinger's Integral 82. The Case of a Single Variable 83. Properties of the Hellinger IntegralChapter IV Metric and Normed Spaces 84. Metric Space 85. The Completion of a Metric Space 86. Operators and Functionals. The Principle of Compressed Mappings 87. Examples 88. Examples of Applications of the Principle of Compressed Mappings 89. Compactness 90. Compactness in C 91. Compactness in Lp 92. Compactness In Lp 93. Functionals on Mutually Compact Sets 94.