Berberian S.K. — Fundamentals of Real Analysis :: Электронная библиотека попечительского совета мехмата МГУ

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Berberian S.K. — Fundamentals of Real Analysis

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Название: Fundamentals of Real Analysis

Автор: Berberian S.K.

Аннотация:

Integration theory and general topology form the core of this textbook for a first-year graduate course in real analysis. After the foundational material in the first chapter (construction of the reals, cardinal and ordinal numbers, Zom's Lemma, and transfinite induction), measure, integral, and topology are introduced and developed as recurrent themes of increasing depth.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 1998

Количество страниц: 479

Добавлена в каталог: 04.04.2008

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Предметный указатель

-algebra      352
-norm      314
-norm      332
$\sigma$ -algebra of subsets      101
$\sigma$ -finite measure on an algebra of sets      369
$\sigma$ -ring of subsets      432
a.e.      157
a.e.primitive      248
a.e.unique      157
Absolute continuity of measures      192 322 440
Absolute continuity of signed measures      440 445
Absolutely continuous function      206 213
Adherence of a subset of a topological space      134
Adherent point      133
Algebra      212
Algebra of sets      101
Almost everywhere      157
Arzela — Young theorem      113
Ascoli’s Theorem      401 407
Axiom of Choice      13
Baire category theorem      304
Baire space      304
Banach algebra      331 352
Banach space      318 461
Base for a topology      279
Basis of a vector space      43
Bijection      10
Bijective function      10
Bilateral sequence      11
Bilinear mapping      344
Boolean ring      432
Boolean ring, of sets      107
Borel function      151 453
Borel sets of $\mathcal{R}$       102
Borel sets of $\mathcal{R}$ , of a topological space      149
Borel space      149
Boundary of a subset of a topological space      137
Boundary point      137
Bounded above      22
Bounded below      22
Bounded function      119 288
Bounded set, in a metric space      298
Bounded set, in a metric space, in a pre-ordered set      22
Bounded variation, function of      202 264
Canonical factorization of a function      20
Cantor set      99
Cantor’s theorem      50
Cardinal arithmetic      52
Cardinal numbers      49
Cardinal numbers, countable      52
Cardinal numbers, finite      52
Cardinal numbers, infinite      52
Cardinal numbers, uncountable      52
Cardinality      49
Cardinality, finite      37
Cardinality, of an ordinal number      68
Cartesian product      5
Cauchy sequence      278
Cauchy’s criterion for convergence      31
Cauchy’s criterion for convergence, for uniform convergence      287 288
Cavalieri’s principle      397
Chain in an ordered set      21
Characteristic function of a subset      10
Choice function      14
Closed ball in a metric space      126
Closed set in a metric space      126
Closed set in a metric space, in a topological space      131
Closure of a subset of a topological space      134
Commutative diagram      20
Compact linear mapping      461
Compact space      275
Compact space, subset      276
Complement of a subset      4
Complete measure space      108 156
Complete measure space, metric space      282
Complete measure space, ordered field      26 33
Complete measure space, seminormed space      461
Completion of a metric space      309
Complex measure      319
Complex measure, numbers      2 118 350
Composite function      10
Composition of functions      10
Composition of functions, of relations      8
Conclusion of a proposition      3
Congruence of integers      17
Continuity at a point      139
Continuous function      139
Continuous function, linear mapping      461
Continuum Hypothesis      58
Continuum, cardinal of      49 57
Contraction of a measure to a subset      389
Contradiction, law of      5
Contrapositive form of a theorem      3
Convergent sequence in $\bar{\matnbb{R}}$       82
Convergent sequence in $\bar{\matnbb{R}}$ , in a metric space      124
Convergent sequence in $\bar{\matnbb{R}}$ , sequence in a topological space      137
Convergent sequence of functions, pointwise      153 286
Converse of a proposition      3
Convolution of functions      458
Coordinate      13
Countable set      38
Countably infinite set      39
Covering      274
De Morgan’s formulas      13
Decreasing net      77
Dedekind cut      32
Dense subset of a topological space      136
Denumerable set      39
Denumerably infinite set      39
Derivates of a function      220 236
Derivative of a function      145
Diagonal      8 17
Diameter of a subset      130 278 298 299
Difference of two subsets      4
Differentiable function      145
Differentiation, term-by-term      296
Dini derivates of a function      220
Dini’s theorem      293
Dirac measure      106
Discrete measure      106
Discrete measure, metric      118
Discrete measure, metric space      118
Discrete measure, topology on a set      131
Distance function      118
Domain of a relation      8
Dominated Convergence Theorem      176
Dual of an order relation      21
Dual space      343
empty function      17
Equi-uniformly continuous set of functions      400
Equicontinuous set of functions      400
Equicontinuous set of functions, at a point      399
Equipollent sets      46
Equivalence class      18
Equivalence relation      9 17
Equivalence relation, Deduced from a function      17
Equivalent metrics      131
Equivalent metrics, propositions      3
Equivalent metrics, signed measures      448
Essential bound      323
Essential supremum of a function      157 323
Essentially bounded function      157 323
Essentially unique      157
Euclidean metric      122
Euclidean metric, space      122
Excluded middle, law of      5
Extended real numbers      73
Extended real numbers, algebraic operations on      74
Exterior measure      97
Exterior of a subset of a topological space      137

Family indexed by a set      11
Fatou’s Lemma      177
Final set of a function      9
Finite character, a set of sets of      43
Finite covering      274
Finite diameter, subset of      130 278
Finite intersection property      275
Finite set      34 38
Finite signed measure      190 319
Finite-dimensional normed spaces, F.Riesz’s characterization      468
Finitely additive set function      112
First category, subset of the      303
First countable topological space      133
Frequently      80
Fubini — Tonelli theorem      385 392
Fubini’s Theorem      387 395
Function      9
Function of bounded variation      202 264
Fundamental sequence of neighborhoods      133
Fundamental theorem of calculus, Lebesgue’s      252
Generated algebra      101
Generated algebra, $\sigma$ -algebra      101
Graph of a function      9
Hahn decomposition of a finite signed measure      192
Hahn — Kolmogorov extension theorem      367
Hausdorff space      146 275
Hausdorff’s maximality principle, after      42
Heine — Borel theorem      273
Hilbert space      345
Holder’s Inequality      120 332
Homeomorphism      140
Hypothesis of a proposition      3
Ideal, order      63
Identity indexing      11
Identity indexing, mapping      10
Identity indexing, relation      8
Image, direct      7
Imaginary part of a complex-valued function      349
Increasing net      77
Indefinite integral      184 207 213 248
Index set      11
Induction, principle of mathematical      23
Inductive partially ordered set      42
Infimum      73
Infinite set      34 38
Initial ordinal of a cardinal number      69
Initial segment of a well-ordered set      60
Initial set of a function      9
Injection      10
Injective function      10
Inner product      344
Inner regularity of Lebesgue measure      105
Insertion mapping      10
Integers      2
Integrable function      167 178 312
Integrable function, simple function      160
integral      167 177 312
Interior measure      97
Interior of a subset of a topological space      137
Interior point      127 132
Intersection of subsets      4 13
Inverse function      16
Inverse function, image      7 11
Isometric mapping      307
Isometry      307
Iterated integral      384
Iterated limits theorem      294
Jordan decomposition of a function of bounded variation      205
Jordan — Hahn decomposition of a signed measure      431
Kernel function      463
l.s.c.      229 232
Lebesgue decomposition of a function of bounded variation      264
Lebesgue decomposition of a function of bounded variation, of signed measures      446 450
Lebesgue measure      96
Lebesgue number of a covering      401
Lebesgue outer measure      87
Lebesgue-integrable function      167
Lebesgue-measurable function      150
Lebesgue-measurable function, set      93
Lebesgue’s criterion for Riemann integrability      268
Lebesgue’s criterion for Riemann integrability, Fundamental theorem of calculus      252
Lebesgue’s criterion for Riemann integrability, singular function      210
Left limit of a function      144
Left-derivative of a function      145
Left-differentiable function      145
Lemma on monotone classes      180
Lexicographic ordering of a product of ordered sets      22 25 45
Lexicographic ordering of a product of ordered sets, reverse of      71
Limit inferior of a function      216 233
Limit inferior of a sequence      80
Limit of a function at a point      142
Limit of a sequence in $\bar{\mathbb{R}}$       82
Limit of a sequence in a metric space      124
Limit of a sequence of functions, pointwise      153 286
Limit ordinal      71
Limit superior of a function      216 233
Limit superior of a sequence      79
Lindelof’s theorem      280
Linear mapping, compact      461
Linear mapping, compact, continuous      461
Linear ordering      21
Lipschitz condition      206 270 328 410
Locally compact space      363
Lower bound      22
Lower derivate of a function      236
Lower semicontinuous function      229 232
Lower semicontinuous regularization of a function      268
Majorant      22
Mapping      9
Mathematical induction, principle of      23
Maximal element in a pre-ordered set      22
Meager subset      303
Mean, convergence in      315
Mean, convergence in, of order p      461
Measurable function      149 311
Measurable function, rectangle      372
Measurable function, space      149
Measurable function, subset      93 95 149 366
Measure on a $\sigma$ -algebra      103
Measure on an algebra of sets      365
Measure space      103
Metric      117
Metric space      118
Metric subspace      138
Metrizable topological space      131
Minimal element in a pre-ordered set      23
Minkowski p-metric      122
Minkowski p-norm      119
Minkowski space      122
Minkowski’s inequality      120 333
Minorant      22
Monotone class of sets      179
Monotone class of sets, lemma on      180
Monotone Convergence Theorem      176
Monotone convergence theorem, set function      112
Monotonic functions      222 253
Mutually singular signed measures      444
Negation of a proposition      2
Negative part of a function      152
Negligent function      213
Negligible set      87 95 156
Neighborhood of a point      128 132
Nested closed sets property      299
Nested intervals, theorem on      32
Net      77
Nonmeager subset      304
Nonmeasurable set, Vitali’s example      110
Nonnegative integers      2
Nonoverlapping intervals      206
Nonzero set of a function      160

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