Measure and integral : an introduction to real analysis / Richard L. Wheeden, Antoni Zygmund.
Series Monographs and textbooks in pure and applied mathematics ; 43Editor: New York : M. Dekker, c1977Descripción: x, 274 p. : il. ; 24 cmISBN: 0824764994Tema(s): Measure theory | Integrals, GeneralizedOtra clasificación: 28-01 (26-01 42-01)Introduction Chapter [1] Preliminaries [1] 1. Points and Sets in Rn [1] 2. Rn as a Metric Space [2] 3. Open and Closed Sets in Rn; Special Sets [5] 4. Compact Sets; the Heine-Borel Theorem [8] 5. Functions [9] 6. Continuous Functions and Transformations [10] 7. The Riemann Integral [11] Exercises [12] Chapter [2] Chapter [3] Functions of Bounded Variation; the Riemann-Stieltjes Integral [15] 1. Functions of Bounded Variation [15] 2. Rectifiable Curves [21] 3. The Riemann-Stieltjes Integral [23] 4. Further Results About Riemann-Stieltjes Integrals [28] Exercises [31] Lebesgue Measure and Outer Measure [33] 1. Lebesgue Outer Measure; the Cantor Set [33] 2. Lebesgue Measurable Sets [37] 3. Two Properties of Lebesgue Measure [40] 4. Characterizations of Measurability [42] 5. Lipschitz Transformations of Rn [44] 6. A Nonmeasurable Set [46] Exercises [47] Chapter 4 Lebesgue Measurable Functions [50] 1. Elementary Properties of Measurable Functions [51] 2. Semicontinuous Functions [55] 3. Properties of Measurable Functions: Egorov’s Theorem and Lusin’s Theorem [56] 4. Convergence in Measure [59] Exercises [61] Chapter 5 The Lebesgue Integral [64] 1. Definition of the Integral of a Nonnegative Function [64] 2. Properties of the Integral [66] 3. The Integral of an Arbitrary Measurable f [71] 4. A Relation Between Riemann-Stieltjes and Lebesgue Integrals; the Lp Spaces, 0 < p < ∞ [76] 5. Riemann and Lebesgue Integrals [83] Exercises [85] Chapter [6] Repeated Integration [87] 1. Fubini’s Theorem [87] 2. Tonelli’s Theorem [91] 3. Applications of Fubini’s Theorem [93] Exercises [96] Chapter [7] Differentiation [98] 1. The Indefinite Integral [98] 2. Lebesgue’s Differentiation Theorem [100] 3. The Vitali Covering Lemma [109] 4. Differentiation of Monotone Functions [111] 5. Absolutely Continuous and Singular Functions [115] 6. Convex Functions [118] Exercises [123] Chapter 8 Lp Classes [125] 1. Definition of Lp [125] 2. Holder’s Inequality; Minkowski’s Inequality [127] 3. Classes lp [130] 4. Banach and Metric Space Properties [131] 5. The Space L2; Orthogonality [135] 6. Fourier Series; Parseval’s Formula [137] 7. Hilbert Spaces [141] Exercises [143] Chapter 9 Approximations of the Identity; Maximal Functions [145] 1. Convolutions [145] 2. Approximations of the Identity [148] 3. The Hardy-Littlewood Maximal Function [155] 4. The Marcinkiewicz Integral [157] Exercises [159] Chapter 10 Abstract Integration [161] 1. Additive Set Functions; Measures [161] 2. Measurable Functions; Integration 3. Absolutely Continuous and Singular Set [167] Functions and Measures [174] 4. The Dual Space of Lp [182] 5. Relative Differentiation of Measures [185] Exercises [190] Chapter 11 Outer Measure; Measure [193] 1. Constructing Measures from Outer Measures [193] 2. Metric Outer Measures [196] 3. Lebesgue-Stieltjes Measure [197] 4. Hausdorff Measure [201] 5. The Carath6odory-Hahn Extension Theorem [204] Exercises [208] Chapter [12] A Few Facts From Harmonic Analysis [211] 1. Trigonometric Fourier Series [211] 2. Theorems about Fourier Coefficients [217] 3. Convergence of S[f] and Ŝ[f] [222] 4. Divergence of Fourier Series [227] 5. Summability of Sequences and Series [229] 6. Summability of S[f] and Ŝ[f] by the Method of the Arithmetic Mean [235] 7. Summability of S[f] by Abel Means [246] 8. Existence of ͞f [250] 9. Properties of ͞f for f ϵ Lp, 1 < p < ∞ [256] 10. Application of Conjugate Functions to Partial Sums of S[f] 259 Exercises [260] Notation [265] Index [267]
| Item type | Home library | Shelving location | Call number | Materials specified | Copy number | Status | Date due | Barcode | Course reserves |
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 28 W561 (Browse shelf) | Available | A-6128 | ||||
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 28 W561 (Browse shelf) | Ej. 2 | Available | A-6331 |
Introduction --
Chapter [1] --
Preliminaries [1] --
1. Points and Sets in Rn [1] --
2. Rn as a Metric Space [2] --
3. Open and Closed Sets in Rn; Special Sets [5] --
4. Compact Sets; the Heine-Borel Theorem [8] --
5. Functions [9] --
6. Continuous Functions and Transformations [10] --
7. The Riemann Integral [11] --
Exercises [12] --
Chapter [2] --
Chapter [3] --
Functions of Bounded Variation; the Riemann-Stieltjes Integral [15] --
1. Functions of Bounded Variation [15] --
2. Rectifiable Curves [21] --
3. The Riemann-Stieltjes Integral [23] --
4. Further Results About Riemann-Stieltjes Integrals [28] --
Exercises [31] --
Lebesgue Measure and Outer Measure [33] --
1. Lebesgue Outer Measure; the Cantor Set [33] --
2. Lebesgue Measurable Sets [37] --
3. Two Properties of Lebesgue Measure [40] --
4. Characterizations of Measurability [42] --
5. Lipschitz Transformations of Rn [44] --
6. A Nonmeasurable Set [46] --
Exercises [47] --
Chapter 4 Lebesgue Measurable Functions [50] --
1. Elementary Properties of Measurable Functions [51] --
2. Semicontinuous Functions [55] --
3. Properties of Measurable Functions: Egorov’s Theorem and Lusin’s Theorem [56] --
4. Convergence in Measure [59] --
Exercises [61] --
Chapter 5 The Lebesgue Integral [64] --
1. Definition of the Integral of a Nonnegative Function [64] --
2. Properties of the Integral [66] --
3. The Integral of an Arbitrary Measurable f [71] --
4. A Relation Between Riemann-Stieltjes and Lebesgue Integrals; the Lp Spaces, 0 < p < ∞ [76] --
5. Riemann and Lebesgue Integrals [83] --
Exercises [85] --
Chapter [6] --
Repeated Integration [87] --
1. Fubini’s Theorem [87] --
2. Tonelli’s Theorem [91] --
3. Applications of Fubini’s Theorem [93] --
Exercises [96] --
Chapter [7] --
Differentiation [98] --
1. The Indefinite Integral [98] --
2. Lebesgue’s Differentiation Theorem [100] --
3. The Vitali Covering Lemma [109] --
4. Differentiation of Monotone Functions [111] --
5. Absolutely Continuous and Singular Functions [115] --
6. Convex Functions [118] --
Exercises [123] --
Chapter 8 Lp Classes [125] --
1. Definition of Lp [125] --
2. Holder’s Inequality; Minkowski’s Inequality [127] --
3. Classes lp [130] --
4. Banach and Metric Space Properties [131] --
5. The Space L2; Orthogonality [135] --
6. Fourier Series; Parseval’s Formula [137] --
7. Hilbert Spaces [141] --
Exercises [143] --
Chapter 9 Approximations of the Identity; Maximal Functions [145] --
1. Convolutions [145] --
2. Approximations of the Identity [148] --
3. The Hardy-Littlewood Maximal Function [155] --
4. The Marcinkiewicz Integral [157] --
Exercises [159] --
Chapter 10 Abstract Integration [161] --
1. Additive Set Functions; Measures [161] --
2. Measurable Functions; Integration --
3. Absolutely Continuous and Singular Set [167] --
Functions and Measures [174] --
4. The Dual Space of Lp [182] --
5. Relative Differentiation of Measures [185] --
Exercises [190] --
Chapter 11 Outer Measure; Measure [193] --
1. Constructing Measures from Outer Measures [193] --
2. Metric Outer Measures [196] --
3. Lebesgue-Stieltjes Measure [197] --
4. Hausdorff Measure [201] --
5. The Carath6odory-Hahn Extension Theorem [204] --
Exercises [208] --
Chapter [12] --
A Few Facts From Harmonic Analysis [211] --
1. Trigonometric Fourier Series [211] --
2. Theorems about Fourier Coefficients [217] --
3. Convergence of S[f] and Ŝ[f] [222] --
4. Divergence of Fourier Series [227] --
5. Summability of Sequences and Series [229] --
6. Summability of S[f] and Ŝ[f] by the Method of the Arithmetic Mean [235] --
7. Summability of S[f] by Abel Means [246] --
8. Existence of ͞f [250] --
9. Properties of ͞f for f ϵ Lp, 1 < p < ∞ [256] --
10. Application of Conjugate Functions to Partial Sums of S[f] 259 Exercises [260] --
Notation [265] --
Index [267] --
MR, 58 #11295
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