Pfeffer W.F., Fulton W. (Ed) — Riemann Approach to Integration: Local Geometric Theory :: Электронная библиотека попечительского совета мехмата МГУ

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Pfeffer W.F., Fulton W. (Ed) — Riemann Approach to Integration: Local Geometric Theory

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Название: Riemann Approach to Integration: Local Geometric Theory

Авторы: Pfeffer W.F., Fulton W. (Ed)

Аннотация:

This book presents a detailed and mostly elementary exposition of the generalized Riemann-Stieltjes integrals discovered by Henstock, Kurzweil, and McShane. Along with the classical results, it contains some recent developments connected with lipeomorphic change of variables and the divergence theorem for discontinuously differentiable vector fields.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 320

Добавлена в каталог: 11.06.2008

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Предметный указатель

$\sigma$ -algebra      48
$\sigma$ -Lipschitz map      203
Almost all      56
Almost derivable      230
Almost differentiable      120 205
Almost everywhere      56
Almost relative to $\mathcal{F}$       251
bi-Lipschitz map      190
Boundary      4 133
Boundary, essential      279
Boundary, reduced      279
Brouwer theorem on domain invariance      262
Brouwer theorem on fixed point      199
Caliber      126 216
Cantor function      96
Cantor set      96
cell      3 134
Cell, dyadic      53
Cell, face of      148 211
Cell, vertex of      211
Change of variables      28 187 202 204 262 277 291
Concassage      203
Convergence of figures      225
Convergence of functions      18
Convergence, dominated      20 113
Convergence, monotone      18 68 113
Countable additivity      48
Countable set      3
Countable subadditivity      40
cube      134
Cube, dyadic      134
Cube, dyadic, fundamental property of      134
density      118 200
Density point      118 200
Derivate      91 174
Derivate, Lebesgue      71 158
Derivate, relative to $\mathcal{F}$       248
Derivate, relative to S      277
Derivate, symmetric      100
Derivative      25
Diameter      133
Diameter, Euclidean      189
Differentiable function      25
Differentiable map      150
Differentiable vector field      151
Dirac length      146
Dirac volume      146
Directional      195
Directional, partial      150
Distance      133
Divergence      151 290
Divergence, mean      233
Divergence, Theorem      152 224 232 233 237 244 274 276 290
Division      81 135 211
Division, cellular      211
Dyadic cell      53
Dyadic cube      134
Dyadic, fundamental property of      134
Enumeration      3
Equilipschitz sequence      271
Essential boundary      279
Essential closure      279
Essential interior      279
Exterior normal      149 274
figure      125 210
Figure, $(\varepsilon, \eta)$ -small      126 216
Figure, regularity of      239
Figure, shape of      239
Flux      149
Flux from BV set      286
Flux from cell      149
Flux from figure      212 213
Flux from solid      274
Function      4
Function of bounded variation      82 168 228
Function,       115 229
Function, $\mathcal{F}AC_*$       249
Function, AC      83 169
Function, additive, associated      4
Function, additive, continuous      213 258 285
Function, additive, of BV sets      285
Function, additive, of figures      126 212
Function, additive, of intervals      4 136
Function, additive, of solids      258
Function, almost derivable      230
Function, almost derivable, relative to $\mathcal{F}$       251
Function, Cantor      96
Function, characteristic      59 156
Function, decreasing      4
Function, derivable      91 175
Function, derivable, relative to $\mathcal{F}$       248
Function, derivable, relative to S      277
Function, differentiable      25
Function, differentiable, almost      120
Function, extended real-valued      38
Function, extended real-valued, continuous      69
Function, gap      30 50 107
Function, increasing      4
Function, integrable, $\mathcal{B}\mathcal{V}-$       289
Function, integrable, $\mathcal{F}-$       243
Function, integrable, $\mathcal{S}-$       275
Function, integrable, $\mathcal{S}\mathcal{F}-$       273
Function, integrable, g-(gage)      219 222
Function, integrable, Henstock — Kurzweil      102 112
Function, integrable, McShane( — Stieltjes)      8 68 143 158 180 182
Function, integrable, P-      102 112
Function, Lipschitz      15 121
Function, measurable      56 156
Function, monotone      4
Function, semicontinuous      69
Function, simple      59
Function, singular      98
Function, variation of      81 168
Gage      117 215
Gage, null set of      117 215
Gradient      151
Hausdorff measure      255
Homeomorphism      190 262
Hyperplane      155
integral      7 9 143
Integral, $\mathcal{B}\mathcal{V}-$       289
Integral, $\mathcal{F}-$       243
Integral, $\mathcal{S}-$       275
Integral, $\mathcal{S}\mathcal{F}-$       273
Integral, conditionally convergent      103
Integral, depending on parameter      23
Integral, descriptive definition of, full      94 97 177
Integral, descriptive definition of, partial      93 115 229 249
Integral, double      161
Integral, g-(gage)      219 222
Integral, generalized Riemann      9
Integral, Henstock — Kurzweil      102 112
Integral, improper      105 225
Integral, indefinite      81 112 222 243
Integral, iterated      161
Integral, Lebesgue      80
Integral, McShane( — Stieltjes)      9 68 143 158 180 182
Integral, P-      102 112
Integration by parts      32 108 251
interval      3 134

Interval, degenerate      3 134
Lebesgue decomposition      99 178
Lebesgue integral      80
Lebesgue length      5
Lebesgue measure      81 168
Lebesgue partition      5 139
Lebesgue volume      138
Lemma, Cousin      6 140
Lemma, Henstock      17
Lemma, Patou      20
Length      5
Length, Dirac      146
Length, Lebesgue      5
Lipeomorphism      190
Lipschitz constant      121 190
Lipschitz function      15 121
Lipschitz map      190
Luzin condition      185
Luzin map      185
Majorant      72 159
MAP      3
Map, $\sigma$ -Lipschitz      203
Map, bi-Lipschitz      190
Map, determinant of      193 199 202 203
Map, diagonal      192
Map, differentiable      150
Map, differentiable, almost      205
Map, differential of      150
Map, empty      3
Map, Lipschitz      190
Map, Luzin      185
Measurable function      56 156
Measurable set      44 154
Measure of set      38 153
Measure on $\sigma$ -algebra      48
Measure, countable additivity of      48
Measure, countable subadditivity of      40
Measure, Hausdorff      255
Measure, in       153
Measure, in R      40
Measure, Lebesgue      81 168
Measure, Lebesgue, Hausdorff evaluation of      188
Measure, metric      40
Measure, monotonicity of      40
Measure, normalization of      40
Measure, outer      40
Measure, quasi-Hausdorff      254
Metric      133
Metric, Euclidean      133
Minorant      72 159
Negligible set      55 156
normal      149 274
Null set      117 215
Overlapping BV sets      285
Overlapping figures      125 210
Overlapping intervals      3 134
Overlapping solids      258
Partition      5 139
Partition of      6 139
Partition of Amod $(\varepsilon, \eta)$       127 216
Partition of E mod $(\varepsilon, \eta;\mathcal{B}\mathcal{V})$       288
Partition of S mod $(\varepsilon, \eta; \mathcal{S})$       275
Partition, $\delta$ -fine      5 117 139 216
Partition, $\mathcal{F}-$       242
Partition, $\mathcal{S}-$       275
Partition, $\varepsilon$ -regular      243
Partition, $\varepsilon$ -shapely      139
Partition, anchored in      38 139
Partition, body of      125 139
Partition, BV-      288
Partition, empty      6
Partition, in      6 139
Partition, Lebesgue      5 139
Partition, P-(Perron)      5 139
Perimeter of bounded set      281
Perimeter of cell      148
Perimeter of figure      125 210 211
Point of density      118 200
Point of dispersion      118 200
Quasi-Hausdorff measure      254
Reduced boundary      279
Regularity of BV set      288
Regularity of figure      239
Rotation      192
Segment      3
Segmentation      41 136
Set of bounded variation (BV)      281
Set of bounded variation (BV), $(\varepsilon, \eta; \mathcal{B}\mathcal{V})$ -small      288
Set, $F_{\sigma}, G_{\delta}$       50
Set, Cantor      96
Set, countable      3
Set, enumeration of      3
Set, measurable      44 154
Set, negligible      55 156
Set, null      117 215
Set, projection of      211
Set, relatively open      187
Set, section of      211
Set, shape of      246
Set, thin      214
SHAPE      139 239 246
Singular function      98
Singular volume      178
Solid      258
Solid $(\varepsilon, \eta, \mathcal{S})$ -small      275
Solid $\mathcal{S}$ -regularity of      274
Stieltjes sum      7 142 218
Test, Caratheodory      52
Test, Cauchy      10
Test, Lebesgue      79
Test, Perron      76 159
Theorem, Brower, on domain invariance      262
Theorem, Brower, on fixed point      199
Theorem, change of variables      28 187 202 204 262 277 291
Theorem, covering      86 170 246
Theorem, divergence      152 224 232 233 237 244 274 276 290
Theorem, dominated convergence      20 113
Theorem, Fubini      165 184
Theorem, fundamental of calculus      28 103 124
Theorem, integration by parts      30 108 251
Theorem, Kirszbraun      190
Theorem, Lebesgue decomposition      99 178
Theorem, mean value      34 109
Theorem, monotone convergence      18 68 113
Theorem, Rademacher      198
Theorem, Stepanoff      123 205
Theorem, Tonelli      165
Theorem, Vitali      87 171 246
Theorem, Vitali — Caratheodory      78
Vector field      149
Vector field, divergence of      151 233
Vector field, fluxing      237
Vertex      211
Vitali cover      87 171 246
Vitali cover, centered      90
Vitali, covering theorem      87 171 246
Volume      138
Volume, Dirac      146
Volume, Lebesgue      138
Volume, singular      178

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