This book features the interplay of two main branches of mathematics: topology and real analysis. The material of the book is largely contained in the research publications of the authors and their students from the past 50 years. Parts of analysis are touched upon in a unique way, for example, Lebesgue measurability, Baire classes of functions, differentiability, $C^n$ and $C^{\infty}$ functions, the Blumberg theorem, bounded variation in the sense of Cesari, and various theorems on Fourier series and generalized bounded variation of a function. Features: Contains new results and complete proofs of some known results for the first time. Demonstrates the wide applicability of certain basic notions and techniques in measure theory and set-theoretic topology. Gives unified treatments of large bodies of research found in the literature.
Author(s): Casper Goffman
Series: Mathematical Surveys and Monographs
Publisher: American Mathematical Society
Year: 1997
Language: English
Pages: 216
Title......Page 1
Erratum......Page 2
Copyright......Page 3
Dedication......Page 4
Contents......Page 5
The one dimensional case......Page 8
Mappings and measures on R^n......Page 9
Fourier series......Page 10
1.1. Equivalence classes......Page 12
1.2. Lebesgue equivalence of sets......Page 13
1.3. Density topology......Page 14
1.4. The Zahorski classes......Page 19
2.1. Characterization......Page 22
2.2. Absolutely measurable functions......Page 24
2.3. Example......Page 28
3.1. Continuous functions of bounded variation......Page 30
3.2. Continuously differentiable functions......Page 36
3.3. The class C^n[0, 1]......Page 39
3.4. Remarks......Page 47
4.1. Properties of derivatives......Page 49
4.2. Characterization of the derivative......Page 53
4.3. Proof of Maximoff's theorem......Page 55
4.4. Approximate derivatives......Page 61
4.5. Remarks......Page 64
5.1. Lebesgue measurability......Page 66
5.2. Length of nonparametric curves......Page 68
5.3. Nonparametric area......Page 73
5.4. Invariance under self-homeomorphisms......Page 76
5.5. Invariance of approximately continuous functions......Page 77
5.6. Remarks......Page 79
6.1. Background......Page 81
6.2. Approximations by homeomorphisms of one-to-one maps......Page 82
6.3. Extensions of homeomorphisms......Page 84
6.4. Measurable one-to-one maps......Page 88
7.1. Preliminaries......Page 93
7.3. Constructions of deformations......Page 95
7.4. Deformation theorem......Page 100
7.5. Remarks......Page 101
8.1. Blumberg's theorem for metric spaces......Page 103
8.2. Non-Blumberg Baire spaces......Page 107
8.3. Homeomorphism analogues......Page 108
9.1. Preliminaries......Page 113
9.2. Uniform convergence......Page 119
9.3. Conjugate functions and the Pal-Bohr theorem......Page 122
9.4. Absolute convergence......Page 126
10.1. Tests for pointwise and uniform convergence......Page 133
10.2. Fourier series of regulated functions......Page 140
10.3. Uniform convergence of Fourier series......Page 151
11.1. Absolutely measurable functions......Page 161
11.2. Convergence of Fourier series after change of variable......Page 166
11.3. Functions of generalized bounded variation......Page 169
11.4. Preservation of the order of magnitude of Fourier coefficients......Page 180
A.1. Baire, Borel and Lebesgue......Page 188
A.2. Lipschitzian functions......Page 190
A.3. Bounded variation......Page 193
A.4. Density topology......Page 196
A.5. Approximately continuous maps into metric spaces......Page 198
A.6. Hausdorff dimension......Page 199
A.7. Hausdorff packing......Page 200
A.9. Schwarz's example......Page 204
A.10. Lebesgue's lower semicontinuous area......Page 205
A.11. Distribution derivatives for one real variable......Page 206
Bibliography......Page 208
Index......Page 213
