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Measure and Integral An Introduction to Real Analysis 2nd Edition by Richard L Wheeden ISBN 978-1498702898

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Measure and Integral An Introduction to Real Analysis Second Edition Wheeden Digital Instant Download

Author(s): Wheeden, Richard L
ISBN(s): 9781498702904, 1498702902
Edition: 2nd ed
File Details: PDF, 4.72 MB
Year: 2015
Language: english
SKU: EB-5144248 Category: Tags: ,

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ISBN 13: 978-1498702898

Author: Richard L Wheeden

Now considered a classic text on the topic, Measure and Integral: An Introduction to Real Analysis provides an introduction to real analysis by first developing the theory of measure and integration in the simple setting of Euclidean space, and then presenting a more general treatment based on abstract notions characterized by axioms and with less geometric content.

Published nearly forty years after the first edition, this long-awaited Second Edition also:

  • Studies the Fourier transform of functions in the spaces L1, L2, and Lp, 1 <p <2
  • Shows the Hilbert transform to be a bounded operator on L2, as an application of the L2 theory of the Fourier transform in the one-dimensional case
  • Covers fractional integration and some topics related to mean oscillation properties of functions, such as the classes of Hölder continuous functions and the space of functions of bounded mean oscillation
  • Derives a subrepresentation formula, which in higher dimensions plays a role roughly similar to the one played by the fundamental theorem of calculus in one dimension
  • Extends the subrepresentation formula derived for smooth functions to functions with a weak gradient
  • Applies the norm estimates derived for fractional integral operators to obtain local and global first-order Poincaré–Sobolev inequalities, including endpoint cases
  • Proves the existence of a tangent plane to the graph of a Lipschitz function of several variables
  • Includes many new exercises not present in the first edition

This widely used and highly respected text for upper-division undergraduate and first-year graduate students of mathematics, statistics, probability, or engineering is revised for a new generation of students and instructors. The book also serves as a handy reference for professional mathematicians.

Table of contents: 

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright Page
  5. Dedication
  6. Table of Contents
  7. Preface to the Second Edition
  8. Preface to the First Edition
  9. Authors
  10. 1. Preliminaries
  11. 1.1 Points and Sets in Rn
  12. 1.1 Rn as a Metric Space
  13. 1.3 Open and Closed Sets in Rn, and Special Sets
  14. 1.4 Compact Sets and the Heine-Borel Theorem
  15. 1.5 Functions
  16. 1.6 Continuous Functions and Transformations
  17. 1.7 The Riemann Integral
  18. Exercises
  19. 2. Functions of Bounded Variation and the Riemann-Stieltjes Integral
  20. 2.1 Functions of Bounded Variation
  21. 2.2 Rectifiable Curves
  22. 2.3 The Riemann-Stieltjes Integral
  23. 2.4 Further Results about Riemann-Stieltjes Integrals
  24. Exercises
  25. 3. Lebesgue Measure and Outer Measure
  26. 3.1 Lebesgue Outer Measure and the Cantor Set
  27. 3.2 Lebesgue Measurable Sets
  28. 3.3 Two Properties of Lebesgue Measure
  29. 3.4 Characterizations of Measurability
  30. 3.5 Lipschitz Transformations of Rn
  31. 3.6 A Nonmeasurable Set.
  32. Exercises
  33. 4. Lebesgue Measurable Functions
  34. 4.1 Elementary Properties of Measurable Functions
  35. 4.2 Semicontinuous Functions
  36. 4.3 Properties of Measurable Functions and Theorems of Egorov and Lusin
  37. 4.4 Convergence in Measure
  38. Exercises
  39. 5. The Lebesgue Integral
  40. 5.1 Definition of the Integral of a Nonnegative Function
  41. 5.2 Properties of the Integral
  42. 5.3 The Integral of an Arbitrary Measurable f
  43. 5.4 Relation between Riemann-Stieltjes and Lebesgue Integrals, and the Lp Spaces, 0<p<∞
  44. 5.5 Riemann and Lebesgue Integrals
  45. Exercises
  46. 6. Repeated Integration
  47. 6.1 Fubini’s Theorem
  48. 6.2 Tonelli’s Theorem
  49. 6.3 Applications of Fubini’s Theorem
  50. Exercises
  51. 7. Differentiation
  52. 7.1 The Indefinite Integral
  53. 7.2 Lebesgue’s Differentiation Theorem
  54. 7.3 Vitali Covering Lemma
  55. 7.4 Differentiation of Monotone Functions
  56. 7.5 Absolutely Continuous and Singular Functions
  57. 7.6 Convex Functions
  58. 7.7 The Differential in Rn
  59. Exercises
  60. 8. Lp Classes
  61. 8.1 Definition of Lp
  62. 8.2 Hölder’s Inequality and Minkowski’s Inequality
  63. 8.3 Classes lp
  64. 8.4 Banach and Metric Space Properties
  65. 8.5 The Space L2 and Orthogonality
  66. 8.6 Fourier Series and Parseval’s Formula
  67. 8.7 Hilbert Spaces
  68. Exercises
  69. 9. Approximations of the Identity and Maximal Functions
  70. 9.1 Convolutions
  71. 9.2 Approximations of the Identity
  72. 9.3 The Hardy-Littlewood Maximal Function
  73. 9.4 The Marcinkiewicz Integral
  74. Exercises
  75. 10. Abstract Integration
  76. 10.1 Additive Set Functions and Measures
  77. 10.2 Measurable Functions and Integration
  78. 10.3 Absolutely Continuous and Singular Set Functions and Measures
  79. 10.4 The Dual Space of Lp
  80. 10.5 Relative Differentiation of Measures
  81. Exercises
  82. 11. Outer Measure and Measure
  83. 11.1 Constructing Measures from Outer Measures
  84. 11.2 Metric Outer Measures
  85. 11.3 Lebesgue-Stieltjes Measure
  86. 11.4 Hausdorff Measure
  87. 11.5 The Carathéodory-Hahn Extension Theorem
  88. Exercises
  89. 12. A Few Facts from Harmonic Analysis
  90. 12.1 Trigonometric Fourier Series
  91. 12.2 Theorems about Fourier Coefficients
  92. 12.3 Convergence of S[ f] and S[ f]
  93. 12.4 Divergence of Fourier Series
  94. 12.5 Summability of Sequences and Series
  95. 12.6 Summability ofS[ f] and S [ f ]by the Method of the Arithmetic Mean
  96. 12.7 Summability of S[ f] by Abel Means
  97. 12.8 Existence of f˜
  98. 12.9 Properties of f~ for f∈Lp,1<p<∞
  99. 12.10 Application of Conjugate Functions to Partial Sums of S[f]
  100. Exercises
  101. 13. The Fourier Transform
  102. 13.1 The Fourier Transform on L1
  103. 13.2 The Fourier Transform on L2
  104. 13.3 The Hilbert Transform on L2
  105. 13.4 The Fourier Transform on Lp,1<p<2
  106. Exercises
  107. 14. Fractional Integration
  108. 14.1 Subrepresentation Formulas and Fractional Integrals
  109. 14.2 L1, L1 Poincaré Estimates, the Subrepresentation Formula, and Hölder Classes
  110. 14.3 Norm Estimates for Iα
  111. 14.4 Exponential Integrability of Iαf
  112. 14.5 Bounded Mean Oscillation
  113. Exercises
  114. 15. Weak Derivatives and Poincaré-Sobolev Estimates
  115. 15.1 Weak Derivatives
  116. 15.2 Smooth Approximation and Sobolev Spaces
  117. 15.3 Poincaré-Sobolev Estimates
  118. Exercises
  119. Notations
  120. Index

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Tags: Richard L Wheeden, Measure and Integral, An Introduction to Real Analysis