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Natanson I.P. — Theory of Functions of a Real Variable. Volume II
Natanson I.P. — Theory of Functions of a Real Variable. Volume II

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Название: Theory of Functions of a Real Variable. Volume II

Автор: Natanson I.P.

Аннотация:

This is a translation of Professor Natanson's Teoriya Funktsii Veshchestvennoi Peremennoi, Chapters X-XVI, XVIII of the second revised Russian edition, 1957; Chapter XVIII forms the content of Chapter XVII of the present volume. Chapter XVII of the Russian edition was omitted as such because the author follows quite closely (except for Appendix VI) the material in the Boron-Hewitt American edition of Chapters I-IX listed as Editor's Appendices and written by Professor Dr. Edwin Hewitt, dealing with the definition and basic properties of arbitrary measurable sets on the real line, the basic properties of measurable functions defined on sets which are not necessarily bounded, the definition of the Lebesgue integral for functions defined on unbounded sets, square-summable functions defined on arbitrary measurable sets on the real line, functions of finite variation on the infinite line (— infty, infty) and the Stieltjes integral of such functions. Appendices I, II, III are the same as in the second Russian edition. Appendix IV, Change of Variable in the Lebesgue Integral, is §5, Chapter IX, 2nd ed. Appendix V, Hausdorff's Theorem, is §5, Chapter XI, 1st ed. Appendix VI, Indefinite Integrals and Absolutely Continuous Set Functions, is §6, Chapter XVII, 2nd ed. Appendix VII, The Role of Russian and Soviet Mathematicians in the Development of the Theory of Functions of a Real Variable, is Chapter XVII, 1st ed.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1960

Количество страниц: 263

Добавлена в каталог: 07.04.2013

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Предметный указатель
(P)-integrable function      159
Abel condition      30
Abel — Poisson summation method      28
Abel’s lemma      29
Abel’s theorem      30
Absolutely continuous set functions      94 246
Abstract definition of integral      169
Addition of order types      111
Additive set function      94
Affine transformation      72
Akhiezer      257
Akilov      257
Aleksandrov      81 145 157 185 249 250 251
Aleksandrov — Looman theorem      185 189
Alephs      122
Alephs, transfinite      122
Analytic sets      249
Antinomy, Burali — Forti      117
Approximate continuity, derivative      157
Approximate continuity, point of      16
Arc, Euler polygonal      210
Arc, length of a curve      225
Arzel — Ascoli theorem      207
Asymptotic derivative      157
Axiom, Zermelo’s      124
Babenko      255
Baire, classes      128
Baire, classification      128
Baire, functions      43 128
Baire’s theorem      143 146
Banach      195
Banach, algebras      256
Banach, spaces      195
Banach’s fixed-point principle      216
Banach’s theorem      217
Bari      54 57 253 254 255
Bari — Menshov theorem      255
Bari’s theorem      253
Bernstein      248 253
Bernstein, singular integral      58
Birnbaum — Orlicz theorem      232
Bochvar      257
Bogolybov      257
Bolzano — Weierstrass theorem      60
BorePs theorem      61
Bounded, set      199
Bounded, solution      222
Burali — Forti antinomy      117
Bykov      253
C-set      250
Cantor      54
Cantor — Baire stationary principle      145
Cantor — Labesgue theorem      34
Cantor’s perfect set      148
Cantor’s theorem      51 52 55
Center of a circle      60
Cesaro summation method      22
Cesro — Fejer summation method      28
Change of variable      234
Characteristic function      152
chebyshev      248
Closed, circle      60
Closed, sets      60 199
Closed, sphere      199
Compact, set      199
Compact, space      199
Compactness      198 202
Comparison of ordinal numbers      115
Comparison of Perron and Lebesgue integrals      165
Complementary in the Young sense      232
Complete metric space      194
Complete, trigonometric system      25
Completely, additive set function      94
Completely, regular system      222
Conditions for compactness      202
Constructive theory of functions      248
Continuum Hypothesis      127
Contractible family      103
Contraction operator      217
Convergence, weak      15
Convergent in itself      193
Convex, downward      38
Convex, functions      11 36 230
Convex, majorant      20
Countably additive set function      94
Covering in the Vitali sense      70
Covering of a set      61
D-point      57
De La Vallee Poussin      58 234
De la Vallee Poussin, Poussin’s theorem      40
De la Vallee Poussin, singular integral      58
Denjoy      54 157
Denjoy integral in the restricted sense      175
Denjoy integral in the wide sense      189
Denjoy — Khinchin integral      189
Denjoy — Perron integral      175
Derivative with respect to a family of measurable sets      103
Derivative, approximate      157
Derivative, asymptotic      157
Derivative, lower      158
Derivative, Schwarz      36
Derivative, second Schwarz      37
Derivative, symmetric      95
Derivative, upper      158
Derived number, second Schwarz      38
Derived number, set      60
Descriptive theory of functions      248
Determinant of a transformation      72
Diameter of a set      103 201
Differentiation of the indefinite integral      100
Dirichlet singular integral      22
Dirichlet’s function      148 149
Distance      60
Distance in a metric space      193
Ditkin      256
Du Bois-Reymond-de la Vallee Poussin theorem      52
e-net      200
Egorov      249
Equicontinuous functions      206
Euler, function      210
Euler, polygonal arc      210
Faddeyev      21 255
Faddeyev’s theorem      20
Family of sets contractible to a point      103
Fatou      58
Fejer singular integral      23
Fejer — Lebesgue theorem      25
Fichtenholz      22 28 92 249 254 255 256 257
First, category set      142
First, class of functions      128 139
First, number class      119
First, Schwarz derivative      36
Fixed point of an operator      217
Fourier series      21 29
Fubini’s Theorem      86
Function, (P)-integrable      159
Function, absolutely continuous set      94 246
Function, Additive set      94
Function, Baire      43 128
Function, characteristic      152
Function, convex      11 36 230
Function, downward      38
Function, Euler      210
Function, induced      143
Function, integrable in the Perron sense      159
Function, lower semi-continuous      150 152
Function, measurable      80
Function, representation of a (by a singular integral)      15
Function, Riemann      50
Function, semi-continuous      149
Function, set      94
Function, Steklov      214
Function, subgraph of a      83
Function, truncated      87
Function, universal      139
Function, upper semi-continuous      150 152
Functional analysis      193
Functions of the first class      128 139
Functions of the second class      128
Gagayev      133 254
Gantmakher      257
Gavurin      257
Gelfand      253 256
General principle of choice      124
Generalizations of the Lebesgue integral      157
Geometric interpretation of the Lebesgue integral      83
Germeier      251
Glivenko      251
Gnedenko      254
Golubev      257
Grinblyum      255
Hake      157 178
Hake’s theorem      178 185
Hausdorff’s theorem      238
Holder’s Inequality      234
Indefinite, integral      100 246
Indefinite, Perron integral      163
Induced function      143
Inequality, Holder’s      234
Inequality, Jensen integral      46
Inequality, summation      44
Inequality, triangle      193
Inequality, Young’s      231
Integral, (generalizations of the)      157
Integral, (in the wide sense)      157 189
Integral, abstract definition of      169
Integral, Bernstein      58
Integral, de la Vallee Poussin      58
Integral, Denjoy (in the restricted sense)      157 175
Integral, Denjoy — Khinchin      157 189
Integral, Denjoy — Perron      157 175
Integral, Dirichlet      22
Integral, Fejer      23
Integral, indefinite      100 246
Integral, KantoroviC      58
Integral, Landau      58
Integral, Lebesgue      83
Integral, Perron      158
Integral, Poisson      29 58
Integral, singular      12
Integral, Weierstrass      20
Interchanging order of integration      90
Interior point      62 199
Invariants under isometries      70
Isolated point      60
Isometry      70
Jensen integral inequality      46
Jensen integral inequality, summation inequality      44
Jordan’s Theorem      226
Kaczmarz      27
Kantorovic      22 28 58 139 250 251 256 257
Kantorovic, singular integral      58
Kavanko      257
Keldysh      250
Kernel      12
Kharshiladze      253 256
Khinchin      157 251 254
Koenig      127
Kolmogorov      27 212 228 250 251 253 254 256
Kolmogorov’s theorem      26 27 215
Korenblyum      256
Kozlov      159 253 255
Krein      249 255 256 257
Kronrod      256
KRYLOV      257
Lacunary, sequence      26
Lacunary, trigonometric series      27
Landau singular integral      58
Lebesgue      58 124 133
Lebesgue, change of variable in the      234
Lebesgue, generalizations of      157
Lebesgue, integral      83
Lebesgue, point      16
Lebesgue’s theorem      13 15 137 141
Legitimate system of classes      169
Lemma, Abel’s      29
Lemma, Natanson’s      16
Lemma, Riemann’s      48
Lemma, Schwarz’s      55
Levin      256 257
Levitan      257
Levi’s theorem      88
Liberman      251
Limit of a sequence      193
Limit, element      198
Limit, inferior      134 149
Limit, point      60 198
Limit, superior      134 149
Linear, space      194
Linear, spaces which are not normed      197
Livenson      250
Looman      157 185
Lower, derivative      158
Lower, semi-continuous function      150 152
Lozinsky      253 256
Luzin      83 249 250 252 253 254 257
Luzin — Denjoy theorem      35
Lyapunov      249
Lyustemik      256
Majorant, convex      20
Maltsev      257
Markov      119 257
Markushevic      257
Mazurkiewicz      245
Measurability as invariant under isometries      70
Measurable, function      80
Measurable, functions of several variables      80
Measurable, hull of a set      95
Measure      64 67 69 75
Measure as invariant under isometries      70
Measure of sections of a plane set      76
Medvedev      59
Menshov      54 57 252 253 254
Method of ordering      107
Metric, space      193
Metric, theory of functions      248
Milman      255 257
Molodshy      124
Mutually complementary in the Young sense      232
Myshkis      256
Natanson      16 58 59 191 253 255
Natanson’s lemma      16
natural order      107
Nemytsky      217 253 256
Nersesov      253
Nikolsky      253
Non-degenerate transformation      72
Non-vacuousness of the Baire classes      133
Norm      195
Normal element of a set      114
Normed linear space      195
Novikov      249 250 257
Nowhere dense set      142
Null class of functions      128
Number class      124
Number class of the first type      119
Number class of the second type      119
Oberfunktion      158
Open, circle      60
Open, sets      62
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