Nonmeasurable Sets and Functions (eBook)

Alexander Kharazishvili (Autor)

eBook Download: PDF

2004 | 1. Auflage
349 Seiten
Elsevier Science (Verlag)
978-0-08-047976-7 (ISBN)

Infos

The book is devoted to various constructions of sets which are nonmeasurable with respect to invariant (more generally, quasi-invariant) measures. Our starting point is the classical Vitali theorem stating the existence of subsets of the real line which are not measurable in the Lebesgue sense. This theorem stimulated the development of the following interesting topics in mathematics:
1. Paradoxical decompositions of sets in finite-dimensional Euclidean spaces,
2. The theory of non-real-valued-measurable cardinals,
3. The theory of invariant (quasi-invariant)
extensions of invariant (quasi-invariant) measures.

These topics are under consideration in the book. The role of nonmeasurable sets (functions) in point set theory and real analysis is underlined and various classes of such sets (functions) are investigated . Among them there are: Vitali sets, Bernstein sets, Sierpinski sets, nontrivial solutions of the Cauchy functional equation, absolutely nonmeasurable sets in uncountable groups, absolutely nonmeasurable additive functions, thick uniform subsets of the plane, small nonmeasurable sets, absolutely negligible sets, etc. The importance of properties of nonmeasurable sets for various aspects of the measure extension problem is shown. It is also demonstrated that there are close relationships between the existence of nonmeasurable sets and some deep questions of axiomatic set theory, infinite combinatorics, set-theoretical topology, general theory of commutative groups. Many open attractive problems are formulated concerning nonmeasurable sets and functions.

? highlights the importance of nonmeasurable sets (functions) for general measure extension problem.
? Deep connections of the topic with set theory, real analysis, infinite combinatorics, group theory and geometry of Euclidean spaces shown and underlined.
? self-contained and accessible for a wide audience of potential readers.
? Each chapter ends with exercises which provide valuable additional information about nonmeasurable sets and functions.
? Numerous open problems and questions.

Inhalt

Cover 1
Contents 5
Preface 7
Chapter 1. The Vitali Theorem 13
Theorem 1 14
Theorem 2 15
Theorem 3 18
Theorem 4 18
Theorem 5 20
Theorem 6 22
Exercises 24
Chapter 2. The Bernstein Construction 29
Theorem 1 31
Theorem 2 33
Theorem 3 34
Theorem 4 36
Theorem 5 37
Exercises 39
Chapter 3. Nonmeasurable Sets Associated with Hamel Bases 47
Theorem 1 47
Theorem 2 48
Lemma 1 49
Lemma 2 50
Theorem 3 50
Theorem 4 51
Theorem 5 52
Theorem 6 59
Theorem 7 61
Exercises 62
Chapter 4. The Fubini Theorem and Nonmeasurable Sets 68
Theorem 1 75
Lemma 1 77
Lemma 2 80
Theorem 2 81
Theorem 3 84
Theorem 4 85
Exercises 87
Chapter 5. Small Nonmeasurable Sets 91
Theorem 1 92
Theorem 2 93
Theorem 3 94
Theorem 4 96
Lemma 1 99
Lemma 2 101
Theorem 5 102
Exercises 105
Chapter 6. Strange Subsets of the Euclidean Plane 114
Theorem 1 115
Lemma 1 117
Theorem 2 118
Theorem 3 119
Lemma 2 123
Lemma 3 123
Lemma 4 126
Theorem 4 127
Exercises 127
Chapter 7. Some Special Constructions of Nonmeasurable Sets 133
Theorem 1 134
Theorem 2 139
Lemma 1 147
Theorem 3 148
Exercises 149
Chapter 8. The Generalized Vitali Construction 157
Lemma 1 159
Lemma 2 160
Lemma 3 161
Lemma 4 163
Lemma 5 164
Theorem 1 165
Theorem 2 169
Exercises 174
Chapter 9. Selectors Associated with Countable Subgroups 175
Lemma 1 179
Lemma 2 179
Lemma 3 180
Lemma 4 180
Lemma 5 181
Lemma 6 181
Lemma 7 181
Theorem 1 183
Theorem 2 184
Theorem 3 185
Exercises 186
Chapter 10. Selectors Associated with Uncountable Subgroups 191
Theorem 1 194
Theorem 2 197
Theorem 3 198
Exercises 203
Chapter 11. Absolutely Nonmeasurable Sets in Groups 207
Lemma 1 208
Lemma 2 215
Lemma 3 216
Theorem 1 220
Lemma 4 222
Lemma 5. 223
Theorem 2 224
Theorem 3 225
Theorem 4 225
Theorem 5 225
Theorem 6 226
Theorem 7 227
Exercises 228
Chapter 12. Ideals Producing Nonmeasurable Unions of Sets 232
Lemma 1 234
Theorem 1 235
Theorem 2 237
Theorem 3 237
Theorem 4 238
Lemma 2 239
Lemma 3 240
Lemma 4 240
Lemma 5 242
Theorem 5 243
Theorem 6 243
Theorem 7 243
Theorem 8 244
Theorem 9 244
Exercises 245
Chapter 13. Measurability Properties of Subgroups of a Given Group 248
Lemma 1 249
Lemma 2 251
Lemma 3 251
Theorem 1 253
Theorem 2 254
Theorem 3 257
Lemma 4 259
Lemma 5 261
Lemma 6 262
Lemma 7 264
Lemma 8 264
Theorem 4 265
Exercises 267
Chapter 14. Groups of Rotations and Nonmeasurable Sets 271
Lemma 1 278
Lemma 2 279
Lemma 3 279
Lemma 4 279
Theorem 1 280
Theorem 2 281
Exercises 283
Chapter 15. Nonmeasurable Sets Associated with Filters 288
Lemma 1 290
Lemma 2 292
Theorem 1 294
Lemma 3 295
Lemma 4 297
Lemma 5 299
Theorem 2 300
Exercises 302
Appendix 1 Logical Aspects of the Existence of Nonmeasurable Sets 306
Theorem 1 314
Theorem 2 316
Appendix 2 Some Facts From the Theory of Commutative Groups 320
Theorem 1 322
Theorem 2 324
Theorem 3 327
Theorem 4 327
Bibliography 329
Subject Index 346

Produkt-Details

• Erscheint lt. Verlag: 29.5.2004
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
• Themenwelt: Technik
• ISBN-10: 0-08-047976-6 / 0080479766
• ISBN-13: 978-0-08-047976-7 / 9780080479767