Homeomorphisms in analysis

Title......Page 2
Erratum......Page 3
Copyright......Page 4
Dedication......Page 5
Contents......Page 6
The one dimensional case......Page 9
Mappings and measures on R^n......Page 10
Fourier series......Page 11
1.1. Equivalence classes......Page 13
1.2. Lebesgue equivalence of sets......Page 14
1.3. Density topology......Page 15
1.4. The Zahorski classes......Page 20
2.1. Characterization......Page 23
2.2. Absolutely measurable functions......Page 25
2.3. Example......Page 29
3.1. Continuous functions of bounded variation......Page 31
3.2. Continuously differentiable functions......Page 37
3.3. The class C^n[0, 1]......Page 40
3.4. Remarks......Page 48
4.1. Properties of derivatives......Page 50
4.2. Characterization of the derivative......Page 54
4.3. Proof of Maximoff\'s theorem......Page 56
4.4. Approximate derivatives......Page 62
4.5. Remarks......Page 65
5.1. Lebesgue measurability......Page 67
5.2. Length of nonparametric curves......Page 69
5.3. Nonparametric area......Page 74
5.4. Invariance under self-homeomorphisms......Page 77
5.5. Invariance of approximately continuous functions......Page 78
5.6. Remarks......Page 80
6.1. Background......Page 82
6.2. Approximations by homeomorphisms of one-to-one maps......Page 83
6.3. Extensions of homeomorphisms......Page 85
6.4. Measurable one-to-one maps......Page 89
7.1. Preliminaries......Page 94
7.3. Constructions of deformations......Page 96
7.4. Deformation theorem......Page 101
7.5. Remarks......Page 102
8.1. Blumberg\'s theorem for metric spaces......Page 104
8.2. Non-Blumberg Baire spaces......Page 108
8.3. Homeomorphism analogues......Page 109
9.1. Preliminaries......Page 114
9.2. Uniform convergence......Page 120
9.3. Conjugate functions and the Pal-Bohr theorem......Page 123
9.4. Absolute convergence......Page 127
10.1. Tests for pointwise and uniform convergence......Page 134
10.2. Fourier series of regulated functions......Page 141
10.3. Uniform convergence of Fourier series......Page 152
11.1. Absolutely measurable functions......Page 162
11.2. Convergence of Fourier series after change of variable......Page 167
11.3. Functions of generalized bounded variation......Page 170
11.4. Preservation of the order of magnitude of Fourier coefficients......Page 181
A.1. Baire, Borel and Lebesgue......Page 189
A.2. Lipschitzian functions......Page 191
A.3. Bounded variation......Page 194
A.4. Density topology......Page 197
A.5. Approximately continuous maps into metric spaces......Page 199
A.6. Hausdorff dimension......Page 200
A.7. Hausdorff packing......Page 201
A.9. Schwarz\'s example......Page 205
A.10. Lebesgue\'s lower semicontinuous area......Page 206
A.11. Distribution derivatives for one real variable......Page 207
Bibliography......Page 209
Index......Page 214