Rogers C.A. — Hausdorff Measures :: Электронная библиотека попечительского совета мехмата МГУ

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Rogers C.A. — Hausdorff Measures

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Название: Hausdorff Measures

Автор: Rogers C.A.

Аннотация:

When originally published, this text was the first general account of Hausdorff measures, a subject that has important applications in many fields of mathematics. The first of the three chapters contains an introduction to measure theory, paying particular attention to the study of non-sigma-finite measures. The second chapter develops the most general aspects of the theory of Hausdorff measures, and the final chapter gives a general survey of applications of Hausdorff measures followed by detailed accounts of two special applications. This new edition has a foreword by Kenneth Falconer outlining the developments in measure theory since this book first appeared. This book is ideal for graduate mathematicians with no previous knowledge of the subject, but experts in the field will also want a copy for their shelves

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1970

Количество страниц: 187

Добавлена в каталог: 17.04.2008

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Предметный указатель

Abstract space      2
Analytic set      67
Approximation from within      25 35 97 99 100 119 121 122
Approximation from without      24 34
Approximation of $\mathscr{G}_{\delta}$ -sets by $\mathscr{F}_{\sigma}$ -sets      100
Approximation, rational      133 139—140
Auxiliary measure      109 112
Baire category      74—77
Baire category theorem      75—77
Base for topological space      61 68
Blaschke's selection theorem      91—94 125
Bordered sets      125
Borel sets      22 33 167
Brownian motion      133
Cantor ternary set      67 73
Cantor — Bendixson Theorem      61—63 125
Capacity      133
Caratheodory outer measure      1 128
Caratheodory's lemma      30—32
Cartesian products sets      130—131
Category theorem, Baire      75—77
Category, Baire      74—77
Closed sets      22
Closed sets, space of      90—93 125
Compact sets      60 63 67 68 97 112 119 121 122 124 127
Comparison of measures      75—84
Complete metric space      62
Concentrated sets      74 77
Condensation, point of      61
Construction of measure      20 21
Construction of measure, by Method I      9
Construction of measure, by Method I, $\mathscr{C}_{\sigma\rho}$ -regularity      24
Construction of measure, by Method I, ubiquity of      11
Construction of measure, by Method II      27
Construction of measure, by Method II, $\mathscr{C}_{\sigma\delta}$ -regularity      34
Construction of measure, by Method II, metric property      30
Continued fractions      133 135—147
Continued fractions, properties of      136
Continuity, absolute      134 148 166
Continuum Hypothesis      74
Convergence, strong      100 110 121
Countably additive measure      2 8 12
Counting measure      2 84
CURVES      132
Cylinder sets      130—131
Decimal expansion      132
Dense sets      61
density      129 130
Derivative, upper h-derivative      151
Diameter      27 50
Difference, set theoretic      3
Dimension, generalized      78
Dimension, topological      132
Distance sets      131
Distance, between sets      90
Distance, separating sets      28 125
Egoroff's theorem      39
Euclidean space, Lebesgue measure in n-dimensional      40—43
Euclidean space, net measure in      102—103
Existence theorems, for Hausdorff measures      58—77
Finite dimensional space, net measure in      104 122
Finite positive measure, sets of      58 63 67 68 102 119 121 122 132
Fractions, continued      133 135—147
Functions      134—135 147—168
Functions, h-continuous and strongly h-continuous      149 159—168
Functions, non-decreasing continuous      147—148
Functions, partial order of      78
H-continuity and strong h-continuity      149 160—168
Hausdorff measures      50—127
Hausdorff measures, applications of      128—168
Hausdorff measures, definitions of      50—51
Hilbert space      134
Increasing sets lemma      90—101
Information theory      132
Intersection of sets      131—132
Irregular sets      129
Jarnik's theorems      135—147
Lebesgue measure      40—43
Lemma, Caratheodory's      30—32
Lemma, increasing sets      90—101 107
Lemma, increasing sets, for $\mu_{\delta}^{h}$ measure      97
Lemma, increasing sets, for net measure      107
Line segments      134
Lines, assigning of measures to sets of      132
Lipschitz condition      149 159—165
Mapping theorem      53—54
Mappings and Hausdorff measures      53
Measurability      3

Measurability of Souslin sets      48
Measurability, criterion for      4 19
Measurable sets      3 4 15 19 20 32 33 39 45—49
Measure      see also "Auxiliary measure" "Hausdorff "Lebesgue "Metric "Net "Non-sigma-finite "Outer "Positive "Pre-measure" "Radon "Regular "Sigma-finite "Zero
Measure in topological space      22—26 43
Measure, comparison of      78—84
Measure, construction of      20 21
Measure, construction of, by Method I      9 11 24
Measure, construction of, by Method II      27 30 34
Measure, examples of      2 3 14 68
Measure, partial order of      22
Measureless sets      132
Metric measures      30—33
Metric measures, in topological space      43
Metric space, complete      62
Metric space, measures in      26—40
Metric space, separable      63
Metric, definition of a      26
Net measures      101—122
Net, definition of      101
Non-sigma-finite measure, sets of      78 83 123—127
Non-sigma-finite measure, subsets of      100 127
Nowhere-dense sets      74
Null sets      4 8 59 79 81
Open sets      22
Open sets, assigning positive measure to      68
Order, partial of functions      78
Order, partial of measures      22
Outer measure      1 128
Packing      134
Partial order      see "Order"
Perfect sets      61
Point of condensation      61
Point, isolated      74
POLYGONS      134
Positive measure, assigning of, to open sets      68
Positive measure, sets of      63 97 100
Positive measure, subsets of      97 99 124
Pre-measure      9
Projection properties      129 130
Radon measure      1 148 150—151
Radon — Nikodym theorem      150 166
Rational approximation      133 139—140
Real numbers, sets of, in terms of expansions into continued fractions      135—147
Regular measure      13 17 19
Regular measure, $\mathscr{R}$ -regular measure      23
Regular sets      129
Separable metric space      63
Separation of sets      4
Separation of sets, positive      28
Sets      see also "Analytic" "Bordered" "Borel" "Cantor "Cartesian "Closed" "Compact" "Concentrated" "Cylinder" "Dense" "Distance" "Irregular" "Lines" "Measurable" "Measureless" "Nowhere-dense" "Null" "Open" "Perfect" "Real "Regular" "Souslin" "Uncountable" "Zero "Finite "Non-sigma-finite" "Positive" "Sigma-field" "Sigma-finite"
Sets, $\mathscr{F}_{\sigma}$ -sets      35 40 50 101
Sets, $\mathscr{G}_{\delta}$ -sets      34 35 40 50
Sets, category of      74 11
Sets, countably additive measure on $\sigma$ -field of      8—9
Sets, decreasing sequence of      16 18 39 100 112 119
Sets, distance between      90
Sets, increasing sequence of      15 17 31 90 97 100 107
Sets, intersection properties of      131—132
Sets, separation of      4
Sets, separation of, positive      28
Sigma-compact      99
Sigma-field      5
Sigma-field of sets, countably additive measure defined on      8
Sigma-finite measure, sets of      78
Souslin operation      44—49
Souslin set      84—90 97 99 102 119 121 124
Souslin set, measurability of      48
Space      see also "Metric space" "Topological "Ultra-metric
Space of closed sets      90—93 125
Space, abstract      2
Space, finite dimensional      104 122
Space, Hilbert      134
Subsets of finite positive measure      122
Subsets of non-sigma-finite measure      100 127
Subsets of positive measure      97 99 124
Surface area      53—58 128
Tangential properties      129 130
Topological space, base for      61 68
Topological space, measures in      22—26
Topological space, metric measures in      43
Ultra-metric space      104—107 122
Ultra-metric space, net measure in      104—107 122
Uncountable sets      67
Vitali argument      133 151—155
Vitali theorem      57
Zero measure, criterion for sets to have      59
Zero measure, sets of      4 8 59 79 81

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