Wheeden R.L., Zygmund A. — Measure and integral. An introduction to real analysis :: Электронная библиотека попечительского совета мехмата МГУ

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Wheeden R.L., Zygmund A. — Measure and integral. An introduction to real analysis

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Название: Measure and integral. An introduction to real analysis

Авторы: Wheeden R.L., Zygmund A.

Аннотация:

The book presupposes that the reader has a feeling for rigor and some knowledge of elementary facts from calculus. Some material which is no doubt familiar to many readers has been included; its inclusion seemed desirable in order to make the presentation clear and self-contained.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1977

Количество страниц: 274

Добавлена в каталог: 07.11.2005

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Предметный указатель

$f_{\sigma}$       6
$G_{\delta}$       6
$L, L^{1}$       72 170 135 126
$L^{2}$       135
$L^{p}$ classes      82 125 173
$l^{\infty}$       126
$o(\phi(\chi))$       150
$\delta$ -function      191
$\sigma$ -algebra      39 161
$\sigma$ -algebra of sets      39 161
$\sigma$ -algebra smallest containing      40
$\sigma$ -finite measure      178
$\sigma$ -finite measure on an algebra      205
$\Sigma$ -measurable      162
$\Sigma$ -measurable function      167
(F.) Riesz — Fischer theorem      139
(M.) Riesz's theorem      259
Abel summability (A-summability)      231 246
Abel summability (A-summability) nontangential      232 248
Abel — Stolz theorem      232
Abel's theorem (on summability)      232
Absolute value      3
Absolutely continuous function of a real variable      115 116
Absolutely continuous measure      180 182
Absolutely continuous set function      99 174 180
Abstract measure and integration      161
Additive set function      98 162
Algebra of sets      205
Almost everywhere (a.e.)      52 167
Approximation of the identity      123 148
Arithmetic means      231 235
Arithmetic means delayed      234
Axiom of choice (Zermelo)      46
Banach space      131
Basis for $L^{2}$       137
Besicovitch covering lemma      185
Bessel's inequality      138 212 218
Bolzano — Weierstrass theorem      13
Borel $\sigma$ -algebra      40 196
Borel measurable function      50
Borel measure      187
Borel set      40 196
Boundary of a set      5
Bounded convergence theorem      76 173
Bounded function      9 126
Bounded linear functional      182
Bounded overlaps      185
Bounded set      5
Bounded variation, function of      15 113 117 221 238
Cantor set      35
Cantor — Lebesgue function      35 116
Caratheodory outer measure      196
Caratheodory — Hahn extension theorem      206
Caratheodory's theorem      43
Cauchy sequence      4 60 132
Cauchy — Schwarz inequality      3
Change of variable      124
Characteristic function      54
Choice, axiom of      46
Closed interval      7
Closed set      5
Closure of a set      5
Compact set      8
Compact support      9
Complement of a set      2
Complement relative      2
Complete (metric space, normed space)      4 131 132
Complete measure space      190
Complete orthogonal system      137 139 212 218
Complex-valued measurable function      61 125
Conditional expectation      192
Conjugate Dirichlet kernel      223
Conjugate exponents      127
Conjugate function      225 239 242 245 250
Conjugate series      217
Continuity absolute      99 115 116 174 180
Continuity at a point      10
Continuity in $L^{p}$       134
Continuity relative to a set      10
Continuity uniform      11
Continuous function (of a real variable)      10
Continuous set function      99
Convergence in measure      59 190
Convergence in norm      131
Convergence of a sequence of points      3
Convergence of a sequence of sets      165
Convergence of Fourier series      139 222 259
Convergence theorem Bounded      76 173
Convergence theorem Lebesgue dominated      71 76 173
Convergence theorem Monotone      66 75 172
Convergence theorem Uniform      75 172
Convergence weak      144
Converse of Holder's inequality      128 191
Convex function      118
Convolution      93 145
Convolution operator      146
Countably additive family of sets      162
Cover of a set      8
Cover of a set in the Vitali sense      109
Cube in $\mathbf{R}^{n}$       7
Curve      21 22
Cylinder set      65
De Morgan's laws      2
Decomposition Hahn      176
Decomposition Jordan      165
Decomposition Lebesgue      178
Decomposition lemma for functions      250
Decomposition of a function of bounded variation      117
Decomposition of a measure      181
Decomposition of a set function      165 178
Decreasing sequence of sets      2
Dense      4 149
Density, point of      107
Derivates      111
Derivative of a convolution      146
Derivative of a function of bounded variation      113
Derivative of a monotone function      111
Derivative of measures      185
Derivative of the indefinite integral      100
Diameter of a set      5 99
Difference of two sets      2
Differentiability of a monotone function      111
Differentiability of the indefinite integral      100
Differentiation of measures, relative      185
Dini number      111
Dini's test      224
Dirichlet function      16 84
Dirichlet kernel      222
Dirichlet problem      152
Dirichlet — Jordan theorem      238
Discontinuity of the first or second kind      10 19
Discrete measure space      162
Dispersion, point of      107
Distance between two points      3
Distance between two sets      5 196
Distance from a point to a set      95
Distance in a Banach space      131
Distribution function      77
Divergence of Fourier series      227
Dominated Convergence Theorem      71 76 173
Dot product      2
Dual space of a normed linear space      182
Dyadic cubes      8
Egorov's theorem      57 167
Empty set      2
Equiconvergent (series)      226
Equiconvergent (series) in the wider sense      226
Equidistributed (equimeasurable) functions      79
ess inf (essential infimum)      143
ess sup (essential supremum)      125

Essentially bounded      126
Euclidean space      1
Even function      214
Even part of a function      223
Extension of a measure on an algebra      206
Exterior measure      33
Fatou's lemma      70 75 173
Fejer kernel      152 236
Fejer's theorem      236
Finite (-valued) function      9
Fourier coefficient      137 212 217
Fourier integral      217
Fourier series      137 212
Fourier transform      97
Fubini's lemma      113
Fubini's theorem      87
Function absolutely continuous      115 116
Function bounded      9 126
Function Cantor — Lebesgue      35 116
Function characteristic      54
Function conjugate      225 239 250
Function continuous      10
Function convex      118
Function in $L^{p}$       82 125 173
Function integrable      72 170
Function Lipschitz      17 95 115
Function maximal      104 155 244
Function measurable      50 167
Function monotone      9 111
Function of bounded variation      15 113 117 221 238
Function semicontinuous      55 197
Function set      98 162
Function simple      54
Function singular      116
Function step      23
Function uniformly continuous      11
Functional, bounded linear      182
Gauss — Weierstrass kernel      152
Gram — Schmidt process      136
Graph Holder's inequality converse of      128 191
Graph of a curve      21
Graph of a function      64
Greatest lower bound      4
Hahn decomposition      176
Hardy — Littlewood maximal function      104 155 244
Hardy's inequality      192
Hardy's theorem      234
Harmonic analysis      214
Harmonic function      151
Harmonic oscillation      214
Hausdorff dimension      209
Hausdorff measure      202
Hausdorff outer measure      202
Heine — Borel theorem      8
Hilbert space      141
Holder's inequality      128 131 173
Image      11
Improper integral      84
Increasing sequence of sets      2
Indefinite integral      98
Indicator function      54
Inf (infimum)      4
Infinite dimensional Hilbert space      141
Inner measure      48
Inner product      2 135 141
Integrable function      72 170
Integral improper      84
Integral in an abstract space      167 170
Integral Lebesgue      64 72
Integral Lebesgue — Stieltjes      200
Integral of a complex-valued function      76 135
Integral principal value      239
Integral Riemann      12 85
Integral Riemann — Stieltjes      23 76
Integration by parts      26 118
Interior of a set      5
Interior point      5
Intersection of sets      2
Interval in $\mathbf{R}^{1}, \mathbf{R}^{n}$       7
Interval in $\mathbf{R}^{1}, \mathbf{R}^{n}$ open      7
Interval in $\mathbf{R}^{1}, \mathbf{R}^{n}$ partly open      7
Inverse image      51
Isolated point      3
Isometric      140
Isometry      140
Jensen's inequality      119 192
Jensen's integral inequality      122
Jordan decomposition      165
Jordan's Theorem      19
Jump discontinuity      10
Kernel      146
Kernel Dirichlet      222
Kernel Dirichlet conjugate      223
Kernel Fejer      152 236
Kernel Gauss — Weierstrass      152
Kernel Poisson      151 247
Least upper bound      4
Lebesgue constant      228
Lebesgue decomposition      178 181
Lebesgue Differentiation Theorem      100
Lebesgue dominated convergence theorem      71 76 173
Lebesgue integrable function      72 170
Lebesgue integral      64 72 167 170
Lebesgue measurable function      50
Lebesgue measurable set      37
Lebesgue measure      37
Lebesgue outer measure      33
Lebesgue point      108 153 239
Lebesgue theorem on summability      240
Lebesgue — Stieltjes integral      200
Lebesgue — Stieltjes measure and outer measure      199
Length of a curve      21 22
Length of a vector      3
liminf of a function      10
liminf of a sequence of real numbers      4
liminf of a sequence of sets      2
Limit from the right or left      10
Limit point of a sequence or a set      3
limsup of a function      10
limsup of a sequence of real numbers      4
limsup of a sequence of sets      2
Linear functional      182
Linear isometry      140
Linearly independent functions      136
Lipschitz condition      16 115
Lipschitz condition of order $\alpha$       221
Lipschitz transformation      44
Locally integrable function      107
Lower Riemann — Stieltjes sum      27
Lower semicontinuous      55 197
Lower variation of a set function      164
Lusin's theorem      58
Marcinkiewicz integral      95 157 252
Marcinkiewicz's Theorem      95 157 252
max (maximum)      4
Maximal function, Hardy — Littlewood      104 155 244
Mean Value Theorem for Integrals      28
Measurable function      50 167
Measurable function complex-valued      61 125
Measurable set      37 162 193
Measure abstract      162 193
Measure Borel      187
Measure Hausdorff      202
Measure inner      48
Measure Lebesgue      37
Measure Lebesgue — Stieltjes      199
Measure of a set      37 193
Measure on $\sigma$ -algebra      162
Measure on an algebra      205
Measure outer      33 193

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