Theory of functions of a real variable / I. P. Natanson ; translated from the Russian by Leo F. Boron ; with the editorial collaboration of, and with annotations by Edwin Hewitt.
Idioma: Inglés Lenguaje original: Ruso Editor: New York : F. Ungar, c1955-[1961]Descripción: 2 v. ; 26 cmTítulos uniformes: Teoriia funktsii veshchestvennoi peremennoi. Inglés Tema(s): Functions of real variablesOtra clasificación: 26-01CONTENTS CHAPTER I. INFINITE SETS [11] 1. Operations on Sets [11] 2. One-to-One Correspondences [15] 3. Denumerable Sets [17] 4. The Power of the Continuum [21] 5. Comparison of Powers [27] Exercises [33] CHAPTER II. POINT SETS [34] 1. Limit Points [34] 2. Closed Sets [36] 3. Interior Points and Open Sets [41] 4. Distance and Separation [44] 5. The Structure of Bounded Open Sets and Bounded Closed Sets [47] 6. Points of Condensation; the Power of a Closed Set [50] Exercises [54] CHAPTER III. MEASURABLE SETS [55] 1. The Measure of a Bounded Open Set [55] 2. The Measure of a Bounded Closed Set [59] 3. The Outer and Inner Measure of a Bounded Set [63] 4. Measurable Sets [66] 5. Measurability and Measure as Invariants under Isometries [71] 6. The Class of Measurable Sets [75] 7. General Remarks on the Problem of Measure [79] 8. Vitali’s Theorem [81] 9. Editor’s Appendix to Chapter III [84] Exercises [88] CHAPTER IV. MEASURABLE FUNCTIONS [89] 1. The Definition and the Simplest Properties of Measurable Functions [89] 2. Further Properties of Measurable Functions [93] 3. Sequences of Measurable Functions. Convergence in Measure [95] 4. The Structure of Measurable Functions [101] 5. Two Theorems of Weierstrass [107] 6. Editor’s Appendix to Chapter IV [112] Exercises [114] CHAPTER V. THE LEBESGUE INTEGRAL OF A BOUNDED FUNCTION [116] 1. Definition of the Lebesgue Integral [116] 2. Fundamental Properties of the Integral [121] 3. Passage to the Limit under the Integral Sign [127] 4. Comparison of Riemann and Lebesgue Integrals [129] 5. Reconstruction of the Primitive Function [133] CHAPTER VI. SUMMABLE FUNCTIONS [136] 1. The Integral of a Non-negative Measurable Function [136] 2. Summable Functions of Arbitrary Sign [143] 3. Passage to the Limit under the Integral Sign [149] 4. Editor’s Appendix to Chapter VI [159] Exercises [162] CHAPTER VII. SQUARE-SUMMABLE FUNCTIONS 1. Fundamental Definitions. Inequalities. Norm [165] 2. Mean Convergence [167] 3. Orthogonal Systems [175] 4. The Space l2 [184] 5. Linearly Independent Systems [192] 6. The Spaces Lp and lp [196] 7. Editor’s Appendix to Chapter VII [200] Exercises [202] CHAPTER VIII. FUNCTIONS OF FINITE VARIATION. THE STIELTJES INTEGRAL 1. Monotonic Functions [204] 2. Mapping of Sets. Differentiation of Monotonic Functions [207] 3. Functions of Finite Variation [215] 4. Helly’s Principle of Choice [220] 5. Continuous Functions of Finite Variation [223] 6. The Stieltjes Integral [227] 7. Passage to the Limit under the Stieltjes Integral Sign [232] 8. Linear Functionals [236] 9. Editor’s Appendix to Chapter VIII [238] Exercises [241] CHAPTER IX. ABSOLUTELY CONTINUOUS FUNCTIONS. THE INDEFINITE LEBESGUE INTEGRAL [243] 1. Absolutely Continuous Functions [243] 2. Differential Properties of Absolutely Continuous Functions [246] 3. Continuous Mappings [248] 4. The Indefinite Lebesgue Integral [252] 5. Points of Density. Approximate Continuity [260] 6. Supplement to the Theory of Functions of Finite Variation and Stieltjes Integrals [263] 7. Reconstruction of the Primitive Function [266] Exercises [270] INDEX [273]
CONTENTS CHAPTER X. SINGULAR INTEGRALS. TRIGONOMETRIC SERIES. CONVEX FUNCTIONS [11] 1. Concept of Singular Integral [11] 2. Representation of a Function by a Singular Integral at a Given Point [15] 3. Application to the Theory of Fourier Series [21] 4. Further Properties of Trigonometric and Fourier Series [29] 5. Schwarz Derivatives and Convex Functions [36] 6. Uniqueness of the Trigonometric Series Expansion of a Function [48] Exercises [57] CHAPTER XI. POINT SETS IN TWO-DIMENSIONAL SPACE [60] 1. Closed Sets [60] 2. Open Sets [62] 3. Theory of Measure of Plane Sets [64] 4. Measurability and Measure as Invariants under Isometries [70] 5. The Relation between the Measure of a Plane Set and the Measures of Its Sections [75] CHAPTER XII. MEASURABLE FUNCTIONS OF SEVERAL VARIABLES AND THEIR INTEGRATION [80] 1. Measurable Functions. Extension of Continuous Functions [80] 2. The Lebesgue Integral and Its Geometric Interpretation [83] 3. Fubini’s Theorem [85] 4. Interchanging the Order of Integration [90] CHAPTER XIII. SET FUNCTIONS AND THEIR APPLICATIONS IN THE THEORY OF INTEGRATION [94] 1. Absolutely Continuous Set Functions [94] 2. The Indefinite Integral and the Differentiation of It [100] 3. Generalization of the Preceding Results [103] CHAPTER XIV. TRANSFINITE NUMBERS [107] 1. Ordered Sets. Order Types [107] 2. Well-Ordered Sets [112] 3. Ordinal Numbers [115] 4. Transfinite Induction [118] 5. The Second Number Class [119] 6. Alephs [122] 7. Zermelo’s Axiom and Theorem [124] CHAPTER XV. THE BAIRE CLASSIFICATION [128] 1. Baire Classes [128] 2. Non-Vacuousness of the Baire Classes [133] 3. Functions of the First Class [139] 4. Semi-Continuous Functions [149] CHAPTER XVI. CERTAIN GENERALIZATIONS OF THE LEBESGUE INTEGRAL [157] 1. Introduction [157] 2. Definition of the Perron Integral [158] 3. Fundamental Properties of the Perron Integral [160] 4. The Indefinite Perron Integral [163] 5. Comparison of the Perron and Lebesgue Integrals [165] 6. Abstract Definition of the Integral and Its Generalizations [169] 7. The Denjoy Integral in the Restricted Sense (i.e. the Denjoy-Perron Integral) [175] 8. H. Hake’s Theorem [178] 9. The P. S. Aleksandrov-H. Looman Theorem [185] 10. The Denjoy Integral in the Wide Sense (i.e. the Denjoy-Khinchin Integral) [189] Exercises [191] CHAPTER XVIII. SOME IDEAS FROM FUNCTIONAL ANALYSIS [193] 1. Metric and, in Particular, Normed Linear Spaces [193] 2. Compactness [198] 3. Conditions for Compactness in Certain Spaces [202] 4. Banach’s “Fixed-Point Principle’’ and Some of Its Applications [216] APPENDICES [225] I. Arc Length of a Curve [225] II. Steinhaus’s Example [228] III. Certain Supplementary Information about Convex Functions [230] IV. Change of Variable in the Lebesgue Integral [234] V. Hausdorff’s Theorem [238] VI. Indefinite Integrals and Absolutely Continuous Set Functions [246] VII. The Role of Russian and Soviet Mathematicians in the Development of the Theory of Functions of a Real Variable [248] INDEX [259]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 26 N272 (Browse shelf) | Vol. I | Available | A-2329 | ||
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 26 N272 (Browse shelf) | Vol. II | Available | A-2323 |
Traducción de: Teoriia funktsii veshchestvennoi peremennoi.
CONTENTS --
CHAPTER I. INFINITE SETS [11] --
1. Operations on Sets [11] --
2. One-to-One Correspondences [15] --
3. Denumerable Sets [17] --
4. The Power of the Continuum [21] --
5. Comparison of Powers [27] --
Exercises [33] --
CHAPTER II. POINT SETS [34] --
1. Limit Points [34] --
2. Closed Sets [36] --
3. Interior Points and Open Sets [41] --
4. Distance and Separation [44] --
5. The Structure of Bounded Open Sets and Bounded Closed Sets [47] --
6. Points of Condensation; the Power of a Closed Set [50] --
Exercises [54] --
CHAPTER III. MEASURABLE SETS [55] --
1. The Measure of a Bounded Open Set [55] --
2. The Measure of a Bounded Closed Set [59] --
3. The Outer and Inner Measure of a Bounded Set [63] --
4. Measurable Sets [66] --
5. Measurability and Measure as Invariants under Isometries [71] --
6. The Class of Measurable Sets [75] --
7. General Remarks on the Problem of Measure [79] --
8. Vitali’s Theorem [81] --
9. Editor’s Appendix to Chapter III [84] --
Exercises [88] --
CHAPTER IV. MEASURABLE FUNCTIONS [89] --
1. The Definition and the Simplest Properties of Measurable Functions [89] --
2. Further Properties of Measurable Functions [93] --
3. Sequences of Measurable Functions. Convergence in Measure [95] --
4. The Structure of Measurable Functions [101] --
5. Two Theorems of Weierstrass [107] --
6. Editor’s Appendix to Chapter IV [112] --
Exercises [114] --
CHAPTER V. THE LEBESGUE INTEGRAL OF A BOUNDED FUNCTION [116] --
1. Definition of the Lebesgue Integral [116] --
2. Fundamental Properties of the Integral [121] --
3. Passage to the Limit under the Integral Sign [127] --
4. Comparison of Riemann and Lebesgue Integrals [129] --
5. Reconstruction of the Primitive Function [133] --
CHAPTER VI. SUMMABLE FUNCTIONS [136] --
1. The Integral of a Non-negative Measurable Function [136] --
2. Summable Functions of Arbitrary Sign [143] --
3. Passage to the Limit under the Integral Sign [149] --
4. Editor’s Appendix to Chapter VI [159] --
Exercises [162] --
CHAPTER VII. SQUARE-SUMMABLE FUNCTIONS --
1. Fundamental Definitions. Inequalities. Norm [165] --
2. Mean Convergence [167] --
3. Orthogonal Systems [175] --
4. The Space l2 [184] --
5. Linearly Independent Systems [192] --
6. The Spaces Lp and lp [196] --
7. Editor’s Appendix to Chapter VII [200] --
Exercises [202] --
CHAPTER VIII. FUNCTIONS OF FINITE VARIATION. THE STIELTJES INTEGRAL --
1. Monotonic Functions [204] --
2. Mapping of Sets. Differentiation of Monotonic Functions [207] --
3. Functions of Finite Variation [215] --
4. Helly’s Principle of Choice [220] --
5. Continuous Functions of Finite Variation [223] --
6. The Stieltjes Integral [227] --
7. Passage to the Limit under the Stieltjes Integral Sign [232] --
8. Linear Functionals [236] --
9. Editor’s Appendix to Chapter VIII [238] --
Exercises [241] --
CHAPTER IX. ABSOLUTELY CONTINUOUS FUNCTIONS. THE INDEFINITE LEBESGUE INTEGRAL [243] --
1. Absolutely Continuous Functions [243] --
2. Differential Properties of Absolutely Continuous Functions [246] --
3. Continuous Mappings [248] --
4. The Indefinite Lebesgue Integral [252] --
5. Points of Density. Approximate Continuity [260] --
6. Supplement to the Theory of Functions of Finite Variation and Stieltjes Integrals [263] --
7. Reconstruction of the Primitive Function [266] --
Exercises [270] --
INDEX [273] --
CONTENTS --
CHAPTER X. SINGULAR INTEGRALS. TRIGONOMETRIC SERIES. --
CONVEX FUNCTIONS [11] --
1. Concept of Singular Integral [11] --
2. Representation of a Function by a Singular Integral at a Given Point [15] --
3. Application to the Theory of Fourier Series [21] --
4. Further Properties of Trigonometric and Fourier Series [29] --
5. Schwarz Derivatives and Convex Functions [36] --
6. Uniqueness of the Trigonometric Series Expansion of a Function [48] --
Exercises [57] --
CHAPTER XI. POINT SETS IN TWO-DIMENSIONAL SPACE [60] --
1. Closed Sets [60] --
2. Open Sets [62] --
3. Theory of Measure of Plane Sets [64] --
4. Measurability and Measure as Invariants under Isometries [70] --
5. The Relation between the Measure of a Plane Set and the Measures of Its Sections [75] --
CHAPTER XII. MEASURABLE FUNCTIONS OF SEVERAL VARIABLES --
AND THEIR INTEGRATION [80] --
1. Measurable Functions. Extension of Continuous Functions [80] --
2. The Lebesgue Integral and Its Geometric Interpretation [83] --
3. Fubini’s Theorem [85] --
4. Interchanging the Order of Integration [90] --
CHAPTER XIII. SET FUNCTIONS AND THEIR APPLICATIONS IN THE THEORY OF INTEGRATION [94] --
1. Absolutely Continuous Set Functions [94] --
2. The Indefinite Integral and the Differentiation of It [100] --
3. Generalization of the Preceding Results [103] --
CHAPTER XIV. TRANSFINITE NUMBERS [107] --
1. Ordered Sets. Order Types [107] --
2. Well-Ordered Sets [112] --
3. Ordinal Numbers [115] --
4. Transfinite Induction [118] --
5. The Second Number Class [119] --
6. Alephs [122] --
7. Zermelo’s Axiom and Theorem [124] --
CHAPTER XV. THE BAIRE CLASSIFICATION [128] --
1. Baire Classes [128] --
2. Non-Vacuousness of the Baire Classes [133] --
3. Functions of the First Class [139] --
4. Semi-Continuous Functions [149] --
CHAPTER XVI. CERTAIN GENERALIZATIONS OF THE LEBESGUE INTEGRAL [157] --
1. Introduction [157] --
2. Definition of the Perron Integral [158] --
3. Fundamental Properties of the Perron Integral [160] --
4. The Indefinite Perron Integral [163] --
5. Comparison of the Perron and Lebesgue Integrals [165] --
6. Abstract Definition of the Integral and Its Generalizations [169] --
7. The Denjoy Integral in the Restricted Sense (i.e. the Denjoy-Perron Integral) [175] --
8. H. Hake’s Theorem [178] --
9. The P. S. Aleksandrov-H. Looman Theorem [185] --
10. The Denjoy Integral in the Wide Sense (i.e. the Denjoy-Khinchin Integral) [189] --
Exercises [191] --
CHAPTER XVIII. SOME IDEAS FROM FUNCTIONAL ANALYSIS [193] --
1. Metric and, in Particular, Normed Linear Spaces [193] --
2. Compactness [198] --
3. Conditions for Compactness in Certain Spaces [202] --
4. Banach’s “Fixed-Point Principle’’ and --
Some of Its Applications [216] --
APPENDICES [225] --
I. Arc Length of a Curve [225] --
II. Steinhaus’s Example [228] --
III. Certain Supplementary Information about Convex Functions [230] --
IV. Change of Variable in the Lebesgue Integral [234] --
V. Hausdorff’s Theorem [238] --
VI. Indefinite Integrals and Absolutely Continuous Set Functions [246] --
VII. The Role of Russian and Soviet Mathematicians in the Development of the Theory of Functions of a Real Variable [248] --
INDEX [259] --
MR, 16,804c (v. 1)
MR, 26 #6309 (v. 2)
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