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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 IIАвтор: 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: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1960Количество страниц: 263Добавлена в каталог: 07.04.2013Операции: Положить на полку |
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(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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