Theory of functions of a real variable, vol. 1,

Title page......Page 1
Foreword to the american edition......Page 3
CONTENTS......Page 5
1. Operations on Sets......Page 9
2. One-to-One Correspondences......Page 13
3. Denumerable Sets......Page 15
4. The Power of the Continuum......Page 19
5. Comparison of Powers......Page 25
Exercises......Page 31
1. Limit Points......Page 32
2. Closed Sets......Page 34
3. Interior Points and Open Sets......Page 39
4. Distance and Separation......Page 42
5. The Structure of Bounded Open Sets and Bounded Closed Sets......Page 45
6. Points of Condensation; the Power of a Closed Set......Page 48
Exercises......Page 52
1. The Measure of a Bounded Open Set......Page 53
2. The Measure of a Bounded Closed Set......Page 57
3. The Outer and Inner Measure of a Bounded Set......Page 61
4. Measurable Sets......Page 64
5. Measurability and Measure as Invariants under Isometries......Page 69
6. The Class of Measurable Sets......Page 73
7. General Remarks on the Problem of Measure......Page 77
8. Vitali's Theorem......Page 79
9. Editor's Appendix to Chapter III......Page 82
Exercises......Page 86
1. The Definition and the Simplest Properties of Measurable Functions......Page 87
2. Further Properties of Measurable Functions......Page 91
3. Sequences of Measurable Functions. Convergence in Measure......Page 93
4. The Structure of Measurable Functions......Page 99
5. Two Theorems of Weierstrass......Page 105
6. Editor's Appendix to Chapter IV......Page 110
Exercises......Page 112
1. Definition of the Lebesgue Integral......Page 114
2. Fundamental Properties of the Integral......Page 119
3. Passage to the Limit under the Integral Sign......Page 125
4. Comparison of Riemann and Lebesgue Integrals......Page 127
5. Reconstruction of the Primitive Function......Page 131
1. The Integral of a Non-negative Measurable Function......Page 134
2. Summable Functions of Arbitrary Sign......Page 141
3. Passage to the Limit under the Integral Sign......Page 147
4. Editor's Appendix to Chapter VI......Page 157
Exercises......Page 160
1. Fundamental Definitions. Inequalities. Norm......Page 163
2. Mean Convergence......Page 165
3. Orthogonal Systems......Page 173
4. The Space l₂......Page 182
5. Linearly Independent Systems......Page 190
6. The Spaces L_p and l_p......Page 194
7. Editor's Appendix to Chapter VII......Page 198
Exercises......Page 200
1. Monotonic Functions......Page 202
2. Mapping of Sets. Differentiation of Monotonic Functions......Page 205
3. Functions of Finite Variation......Page 213
4. Helly's Principle of Choice......Page 218
5. Continuous Functions of Finite Variation......Page 221
6. The Stieltjes Integral......Page 225
7. Passage to the Limit under the Stieltjes Integral Sign......Page 230
8. Linear Functionals......Page 234
9. Editor's Appendix to Chapter VIII......Page 236
Exercises......Page 239
1. Absolutely Continuous Functions......Page 241
2. Differential Properties of Absolutely Continuous Functions......Page 244
3. Continuous Mappings......Page 246
4. The Indefinite Lebesgue Integral......Page 250
5. Points of Density. Approximate Continuity......Page 258
6. Supplement to the Theory of Functions of Finite Variation and Stieltjes Integrals......Page 261
7. Reconstruction of the Primitive Function......Page 264
Exercises......Page 268
INDEX......Page 271