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De Barra G — Measure theory and integration
De Barra G — Measure theory and integration

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Название: Measure theory and integration

Автор: De Barra G

Аннотация:

This updated and introductory text approaches integration via measure as opposed to measure via integration, an approach which makes it easier to grasp the subject. Apart from its central importance to pure mathematics, the material is also relevant to applied mathematics and probability, with proof of the mathematics set out clearly and in considerable detail. Numerous worked examples necessary for teaching and learning at undergraduate level constitute a strong feature of the book, and after studying statements of results of the theorems, students should be able to attempt the 300 problem exercises which test comprehension, for which detailed solutions are provided. The book stems from a long-running successful course and presents the knowledge and experience of Dr. de Barra who has long taught and researched measure theory in London University. This 2nd edition has been updated by the attachment of Afternotes indicating how the subject has developed from material in the text, and misprints from the original have now been corrected. The only pre-requisite is a first course in analysis, and what little topology required is developed within the text.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2000

Количество страниц: 236

Добавлена в каталог: 11.02.2014

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$F_{\sigma}$-sequence      18 32 36 205
$G_{\delta}$-sequence      18 32 36 205
$l^{p}$-norm      109 110
$L^{p}$-space      109 110
$\sigma$-finite      94 98 139 149 171 179 181
$\sigma$-ring      93
Absolute convergence      21 64
Absolutely continuous functions      160 163 228
Absolutely continuous functions, measure      139 161
Algebra      30
Algebra, $\sigma$-algebra      30
Almost everywhere (a.e.)      40 104
Almost uniform convergence      125 128 132
Almost uniformly fundamental      127
Angular measure      187
Approximating measure      45
Arithmetic-geometric mean inequality      114
Axiom of Choice      17 42
Borel measurable      40 165 169 187 211 233
Borel measure      102 158
Borel set      32 43 98 101 102 187 203 211 228
Bounded linear functional      148 174 175
BV [a, b]      81 160 163
Cantor set      24 37 157 202 227
Cantor's function      43 45 196 203
Cantor-like sets      23 26 50 200 202 203
cardinal number      22 203
Cartesian product      15 176
Cauchy sequence      20 118 122
Change of variable      167 234
Characteristic function      22 39
Closure      17
Complement      15
Complete measure      94 101 142
Complete metric space      20 118 124
Completion of a measure      100 102 185
Complex-valued functions      189
Convergence in measure      121 128 131 221
Convergence in the mean      123 128 131 132 221 229
Convergence in the mean of order p      123 128 131 132 223
Convergence, absolute      21 64
Convergence, almost everywhere      118 125 128
Convergence, almost uniform      125 128 132
Convergence, uniform      125 128 131 132 229 235
Convergence, uniform a.e.      125 128
Convex function      5 111 163 215
Convolution      191
Countable set      22
Countable subadditivity      29 95
Countably additive      31 95
De Morgan's laws      15
Dense      17
Density of a set      35
Derivates      77 209
Differentiable      65 87 111
Distance (between sets)      43
Distribution function      156 158
Domain      21
Dual space      148 151 152 227
Egorov's theorem      126 222
Elementary set      176
Equipotent      22
Equivalence class      17
Equivalence class, relation      16
Essential infimum (ess inf)      41 104
Essential supremum (ess sup)      40 104
Essentially bounded      41 104
Euclidean space      16
Extended real numbers      37
Extension of a function      21
Extension of a measure      95
Fatou's lemma      57 58 60 63 88 105 119 123 204 205 208 213
Fourier transform      192
Fubini's theorem      182
Function of bounded variation      81 160 163
Function, Borel      40 165 169 205 228 233
Function, Cantor's      43 45 196 203
Function, characteristic      22
Function, composite      21
Function, convex      5 111 163 215
Function, distribution      156
Function, integrable      61 106 164
Function, Lebesgue's      25 26 78 157 159 163 196 211 234
Function, measurable      38 93 103 169
Function, non-differentiable      79
Function, simple      54
Function, step      22
Function, strictly concave      52
Function, subadditive      30 201
Fundamental in measure      122
Fundamental sequences      121
Generated algebra      32
Generated algebra, $\sigma$-algebra      32
Generated algebra, $\sigma$-ring      94
Generated algebra, ring      94
Hahn decomposition      133 136 137 138 141 223 224 225
Hausdorff measure      45 52 99 158 203
Hausdorff measure, dimension      50 53 159 203
Hausdorff measure, measure function      45 158
Hausdorff measure, outer measure      45
Heine — Borel theorem      18
Helly's theorem      158
Hereditary      94
Hoelders inequality      115 148 150 151 216 217 223 235
Identity mapping      21
Indefinite integral      87 169 171
Induction      16
Infimum      18
Integrable      61 106 164
integral      54 105 106
Integration by parts      65 163 164
Intervals      27
Inversion theorem      195
Iterated limit      20
Iterated limit, integral      182 231 232 233
Jensen's inequality      113 195
Jordan decomposition      137 139 225
Laplace transform      191
Lebesgue decomposition      146 147
Lebesgue dominated convergence theorem      63 107 123
Lebesgue function      25 26 78 157 159 163 196 211 234
Lebesgue integral      54 55
Lebesgue measurable functions      38
Lebesgue measurable set      30 185
Lebesgue measure      31 94 185
Lebesgue monotone convergence theorem      57 105
Lebesgue outer measure      27
Lebesgue set      90 91
Lebesgue — Stieltjes measure      156
Lebesgues Differentiation Theorem      84 85
Limit, lower      18
Limit, one-sided      19
Limit, point      18
Limit, upper      18
Lindeloef's theorem in $\mathbf{R}$      23 84
Lindeloef's theorem in $\mathbf{R}^{n}$      23
Linear functional      148 172
Lipshitz condition      163 228
Mean fundamental sequence      147
Measurable      30 93 97
Measurable function      38 93 103 169
Measurable rectangle      176
Measurable space      102
Measure      31 47 94 153 156 157 179
Measure space      102
Metric      17 116 118 124 125 221
Metric outer measure      49
Minkowski's inequality      115 118 218 219 226
Modulus of continuity      52 159
Monotone class      177 180
Mutually singular measures      137
Negative part of a function      61
Negative set      134
Negative variation of a function      81 164
Non-differentiable function      79
Non-measurable set      42 179 203
Norm      109 110 148 172 226 227
Normed vector space      147
Null set      134
O, o notation      20
One-to-one mapping      21
open      17
Ordinals      182
Ordinate set      184
Outer measure      27 45 94
Parseval's theorem      193 196
Partitions      71
Plancherel transform      194
Positive linear functional      172
Positive part of a function      61
Positive set      134
Positive variation of a function      81 164
primitive      156 157 159
Principle of Finite Induction      16
Product measure      181
Product of measurable spaces      177
Product space      176 185
Pseudometric      17 95 211
Radon — Nikodym derivative      143 167
Radon — Nikodym theorem      139
RANGE      21
rectangle      176
Rectifiable      82
Reflexive space      227
Regular measure      35
Relative topology      18
Riemann integrable      71
Riemann integral      55 71 208
Riemann — Lebesgue lemma      75 235
Riesz Representation Theorem for $L^{1}$      151
Riesz Representation Theorem for $L^{p}$      148
Riesz Representation Theorem for C (I)      172
Ring      93
Schwarz inequality      15 216 218
Sections of functions      179
Sections of sets      178
Sequence, Cauchy      20
Sequence, closed      17
Sequence, compact (closed and bounded)      18 51 159
Sequence, dense      17
Sequence, double      20
Sequence, Lebesgue      90 91
Sequence, measurable      30 93 97
Sequence, monotone decreasing      20
Sequence, monotone increasing      19
Sequence, non-measurable      42 179 203
Sequence, nowhere dense      17 201
Sequence, open      17
Sequence, perfect      17 25
Series, absolutely convergent      21 64
Set, Borel      32 43 98 101 102 187 203 211 228
Sgn x      22
Signed measure      133
Simple function      54
Strictly convex      111
Subadditive      29 95
Support of a measure      157
Supremum      18
Symmetric difference      15
Tchebychev's inequality      107
topology      17
Total variation of a function      81
Total variation of a measure      138
Translation invariance      28 46 186
Uniform convergence      125 128 131 132 229 235
Uniformly fundamental      221
union      15
Unit sphere      187
Variation (of a function), bounded      81
Variation (of a function), negative      81
Variation (of a function), positive      81
Variation (of a function), total      81
Variation (of a measure), total      138
Weakly convergent      120
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