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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.
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Рубрика: Математика /
Статус предметного указателя: Готов указатель с номерами страниц
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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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