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Kharazishvili A.B. — Nonmeasurable Sets and Functions
Kharazishvili A.B. — Nonmeasurable Sets and Functions

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Название: Nonmeasurable Sets and Functions

Автор: Kharazishvili A.B.


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 extensionproblem.
• 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.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 352

Добавлена в каталог: 13.07.2008

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Предметный указатель
(n - n)-correspondence      5
Absolutely negligible set      117
Absolutely nonmeasurable function      84
Absolutely nonmeasurable set      125
Admissible $\sigma$-algebra of sets      67
Admissible binary relation      285
Admissible family of straight lines      111
Admissible functional      133
Admissible group of transformations      109
Admissible transfinite matrix      248
Almost disjoint family of sets      32
Almost invariant set      106
Amenable group      262
Analytic curve      105
Analytic manifold      105
Analytic set      55
Aronszajn tree      143
Axiom of Choice      2
Axiom of Dependent Choice      4
Baire $\sigma$-algebra      130
Baire property      1
Baire topological space      16
Banach theorem      6
Banach — Kuratowski matrix      133
Banach — Kuratowski — Pettis theorem      3
Banach — Tarski paradox      6
Bernstein set      17
Binary relation      1
Borel $\sigma$-algebra      13
Borel base of an ideal      21
Borel equivalence relation      286
Borel subset of a topological space      6
Branch of a tree      140
Cantor discontinuum      18
Cantor — Bernstein theorem      6
Caratheodory conditions      100
Cauchy functional equation      35
Classical Lebesgue measure on the real line      2
Commutative divisible group      242
Commutative injective group      312
Commutative projective group      309
Complete Boolean Algebra      29
Constructible universe      191
Continuous measure      11
Continuous selector      98
Continuum Hypothesis      11
Convex hull of a set      117
Convex polygon      118
Convexly independent set      117
Countable chain condition      15
Countable form of the Axiom of Choice      4
Cyclic group      203
Density point      16
Density topology      16
Dieudonne measure      29
Diffused measure      11
Direct sum of groups      203
Discrete family of sets in a metric space      226
Discrete group      179
Equivalence relation      1
Extension of the Lebesgue measure      30
Filter in a partially ordered set      300
Finitely additive measure      259
First category set      3
Free group      261
Fubini theorem      56
G-measure      145
Generalized Luzin set      12
Generalized Sierpinski set      12
Group of rotations      218
Group of transformations of a set      46
Haar measure on a locally compact topological group      14
Hahn — Banach theorem      272
Hall theorem      7
Hamel basis of the real line      35
Height of a tree      140
Homogeneous covering      118
Homomorphic image of a measure      197
Inaccessible cardinal number      139
Independent rotations      261
Independent set with respect to a relation      51
Independent set with respect to a set-valued mapping      62
Inductive limit of a family of groups      311
Invariant measure      3
Invariant partition      255
Isodyne topological space      32
Jensen inequality      53
Jordan curve      106
Koenig lemma      140
Kolmogorov Extension Theorem      31
Kuratowski — Ulam theorem      29
Kuratowski — Zorn Lemma      38
Lavrentiev theorem      54
Level of a tree      140
Lindeloef topological space      93
Locally compact topological group      14
Locally finite set      122
Lower measurable set-valued mapping      227
Lower semi-continuous function      101
Luzin set      82
Mackey theorem      239
Marczewski $\sigma$-ideal      88
Marczewski set      85
Martin's Axiom      12
Massive set      24
Mazurkiewicz set      102
Measurable partition      13
Measurable selector      98
Metrically transitive measure      10
Montgomery lemma      227
Natural number      2
Negligible set      117
Nonstationary set      29
Orthogonal $\sigma$-ideals of sets      82
Orthogonal transformation of an euclidean space      261
P-closed set      111
Partial function      25
Partial selector of a family of sets      164
Partially ordered set      139
Perfect measure      31
Perfect Subset Property      223
Point-finite family of sets      223
Polish topological space      6
Product group      50
Product measure      47
Projective base of an ideal      221
Projective set      188
Quasi-cyclic group      242
Quasi-invariant measure      165
Quasicompact topological space      7
Radon measure      12
Ramsey theorem      118
Rapid filter      280
Real-valued measurable cardinal      69
Regular cardinal number      139
Resolvable topological space      32
Selector of a family of sets      1
Semifinite measure      149
Set of full measure      24
Set of points in general position      103
Set of Vitali type      2
Set-valued mapping      6
Sierpinski partition      57
Sierpinski set      79
Sierpinski topological space      96
Sierpinski — Erdoes Duality Principle      82
Sierpinski — Zygmund function      34
Small nonmeasurable set      79
Solvable group      196
Stable set in a group      126
Standard Borel measure on the real line      13
Stationary set      29
Steinhaus property      15
Step-function      232
Stochastically independent family of measurable sets      278
Stone — Weierstrass theorem      129
Strongly inaccessible cardinal number      289
Summable family of sets      223
Sup-measurable function      99
Suslin condition      15
Symmetric measure      48
Thick set      24
Totally imperfect set      17
TREE      139
Tree property      140
Tychonoff theorem      7
U-set      33
Ulam transfinite matrix      11
Ultrafilter      123
Uniform set      110
Uniqueness property for invariant measures      46
Universal measure zero space      65
Upper measurable set-valued mapping      227
Vietoris topology      51
Vitali partition      1
Vitali property      146
Vitali set      2
Vitali theorem      1
Von Neumann topology      16
Weak Vitali property      146
Weakly measurable mapping      124
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