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 improved clarity. As before, the strength of the text lies in presenting the student with the difficulties which led to the development of the theory and, whenever possi­ ble, giving the student the tools to overcome those difficulties for himself or herself. Another proverb: Give me a fish, I eat for a day. Teach me to fish, I eat for a lifetime. Soo Bong Chae March 1994 Preface to the First Edition This book was developed from lectures in a course at New College and should be accessible to advanced undergraduate and beginning graduate students. The prerequisites are an understanding of introductory calculus and the ability to comprehend "e-I) arguments. " The study of abstract measure and integration theory has been in vogue for more than two decades in American universities since the publication of Measure Theory by P. R. Halmos (1950). There are, however, very few ele­ mentary texts from which the interested reader with a calculus background can learn the underlying theory in a form that immediately lends itself to an understanding of the subject. This book is meant to be on a level between calculus and abstract integration theory for students of mathematics and physics.

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. Hilbert Spaces.- 6. The Riesz-Fischer Theorem.- 7. Orthonormalization.- 8. Completeness of the Trigonometric System.- 9. Isoperimetric Problem.- 10. Remarks on Fourier Series.- Appendix The Development of the Notion of the Integral by Henri Lebesgue.- Notation.