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| 001 | ocn162130131 | ||
| 003 | OCoLC | ||
| 005 | 20141103172227.0 | ||
| 006 | m o d | ||
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| 008 | 070802s2004 ne ob 001 0 eng d | ||
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| 020 | _a9780444516268 | ||
| 020 | _a0444516263 | ||
| 020 | _a1423741846 (electronic bk.) | ||
| 020 | _a9781423741848 (electronic bk.) | ||
| 020 | _a0080479766 (electronic bk.) | ||
| 020 | _a9780080479767 (electronic bk.) | ||
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| 037 |
_a107953:107995 _bElsevier Science & Technology _nhttp://www.sciencedirect.com |
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| 050 | 4 |
_aQA312 _b.K526 2004eb |
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| 082 | 0 | 4 |
_a515/.42 _222 |
| 049 | _aTEFA | ||
| 100 | 1 | _aKharazishvili, A. B. | |
| 245 | 1 | 0 |
_aNonmeasurable sets and functions _h[electronic resource] / _cA.B. Kharazishvili. |
| 250 | _a1st ed. | ||
| 260 |
_aAmsterdam ; _aBoston : _bElsevier, _c2004. |
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| 300 | _a1 online resource (xi, 337 p.) | ||
| 490 | 1 |
_aNorth-Holland mathematics studies, _x0304-0208 ; _v195 |
|
| 520 | _aThe 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. | ||
| 505 | 0 | _aContents -- Preface. -- 1. The Vitali theorem. -- 2. The Bernstein construction. -- 3. Nonmeasurable sets associated with Hamel bases. -- 4. The Fubini theorem and nonmeasurable sets. -- 5. Small nonmeasurable sets. -- 6. Strange subsets of the Euclidean plane. -- 7. Some special constructions of nonmeasurable sets. -- 8. The Generalized Vitali construction. -- 9. Selectors associated with countable subgroups. -- 10. Selectors associated with uncountable subgroups. -- 11. Absolutely nonmeasurable sets in groups. -- 12. Ideals producing nonmeasurable unions of sets. -- 13. Measurability properties of subgroups of a given group. -- 14. Groups of rotations and nonmeasurable sets. -- 15. Nonmeasurable sets associated with filters. -- Appendix 1: Logical aspects of the existence of nonmeasurable sets. -- Appendix 2: Some facts from the theory of commutative groups. | |
| 504 | _aIncludes bibliographical references (p. 317-333) and index. | ||
| 588 | _aDescription based on print version record. | ||
| 650 | 0 | _aMeasure theory. | |
| 650 | 0 | _aFunctional analysis. | |
| 650 | 0 | _aSet theory. | |
| 650 | 7 |
_aMATHEMATICS _xCalculus. _2bisacsh |
|
| 650 | 7 |
_aMATHEMATICS _xMathematical Analysis. _2bisacsh |
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| 650 | 7 |
_aFunctional analysis. _2fast _0(OCoLC)fst00936061 |
|
| 650 | 7 |
_aMeasure theory. _2fast _0(OCoLC)fst01013175 |
|
| 650 | 7 |
_aSet theory. _2fast _0(OCoLC)fst01113587 |
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| 655 | 4 | _aElectronic books. | |
| 776 | 0 | 8 |
_iPrint version: _aKharazishvili, A.B. _tNonmeasurable sets and functions. _b1st ed. _dAmsterdam ; Boston : Elsevier, 2004 _z0444516263 _z9780444516268 _w(DLC) 2004049706 _w(OCoLC)55078412 |
| 830 | 0 |
_aNorth-Holland mathematics studies ; _v195. _x0304-0208 |
|
| 856 | 4 | 0 |
_3ScienceDirect _uhttp://www.sciencedirect.com/science/book/9780444516268 |
| 856 | 4 |
_uhttp://www.sciencedirect.com/science/publication?issn=03040208&volume=195 _3Volltext |
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_aIngram Digital eBook Collection _bIDEB _n100888 |
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| 938 |
_aEBSCOhost _bEBSC _n132245 |
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| 999 |
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