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Lebesgue Integration

Responses from colleagues and students concerning the first edition indicate that the text still answers a pedagogical need which is not addressed by other texts. There are no major changes in this edition. Several proofs have been tightened, and the exposition has been modified in minor ways for im...

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Bibliographic Details
Main Author: Chae, Soo B.
Format: eBook
Language:English
Published: New York, NY Springer New York 1995, 1995
Edition:2nd ed. 1995
Series:Universitext
Subjects:
Functions Of Real Variables
Mathematical Analysis
Analysis
Real Functions
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 10. Remarks on Fourier Series
  • Appendix The Development of the Notion of the Integral by Henri Lebesgue
  • Notation
  • Zero Preliminaries
  • 1. Sets
  • 2. Relations
  • 3. Countable Sets
  • 4. Real Numbers
  • 5. Topological Concepts in ?
  • 6. Continuous Functions
  • 7. Metric Spaces
  • I The Rieman Integral
  • 1. The Cauchy Integral
  • 2. Fourier Series and Dirichlet’s Conditions
  • 3. The Riemann Integral
  • 4. Sets of Measure Zero
  • 5. Existence of the Riemann Integral
  • 6. Deficiencies of the Riemann Integral
  • II The Lebesgue Integral: Riesz Method
  • 1. Step Functions and Their Integrals
  • 2. Two Fundamental Lemmas
  • 3. The Class L+
  • 4. The Lebesgue Integral
  • 5. The Beppo Levi Theorem—Monotone Convergence Theorem
  • 6. The Lebesgue Theorem—Dominated Convergence Theorem
  • 7. The Space L1
  • Henri Lebesgue
  • Frigyes Riesz
  • III Lebesgue Measure
  • 1. Measurable Functions
  • 2. Lebesgue Measure
  • 3. ?-Algebras and Borel Sets
  • 4. Nonmeasurable Sets
  • 5. Structure of Measurable Sets
  • 6. More About Measurable Functions
  • 7. Egoroff’s Theorem
  • 8. Steinhaus’ Theorem
  • 9. The Cauchy Functional Equation
  • 10. Lebesgue Outer and Inner Measures
  • IV Generalizations
  • 1. The Integral on Measurable Sets
  • 2. The Integral on Infinite Intervals
  • 3. Lebesgue Measure on ?
  • 4. Finite Additive Measure: The Banach Measure Problem
  • 5. The Double Lebesgue Integral and the Fubini Theorem
  • 6. The Complex Integral
  • V Differentiation and the Fundamental Theorem of Calculus
  • 1. Nowhere Differentiable Functions
  • 2. The Dini Derivatives
  • 3. The Rising Sun Lemma and Differentiability of Monotone Functions
  • 4. Functions of Bounded Variation
  • 5. Absolute Continuity
  • 6. The Fundamental Theorem of Calculus
  • VI The LP Spaces and the Riesz-Fischer Theorem
  • 1. The LP Spaces (1 ? p < ?)
  • 2. Approximations by Continuous Functions
  • 3. The Space L?
  • 4. The lp Spaces (1 ? p ? ?)
  • 5. HilbertSpaces
  • 6. The Riesz-Fischer Theorem
  • 7. Orthonormalization
  • 8. Completeness of the Trigonometric System
  • 9. Isoperimetric Problem

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