Lebesgue Integration

Lebesgue Integration presents a logical development of the basic concepts of Lebesgue's integration theorems, proceeding from the study of topological concepts on the real line. This text is intended for advanced undergraduates and beginning graduate students.

Klappentext

The fundamental concept of Lebesgue integration provides a basis for approaching theoretical and applied problems in mathematics, engineering, physics and many other sciences.Lebesgue Integration presents a logical development of the basic concepts of Lebesgue's integration theorems, proceeding from the study of topological concepts on the real line. This text is intended for advanced undergraduates and beginning graduate students and is also suitable for self-study by students preparing for the masters or doctoral preliminary examination.

Inhalt
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 TheoremMonotone Convergence Theorem.- 6. The Lebesgue TheoremDominated 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.- 10. Remarks on Fourier Series.- Appendix The Development of the Notion of the Integral by Henri Lebesgue.- Notation.