Author | Chae, Soo Bong. author |
---|---|
Title | Lebesgue Integration [electronic resource] / by Soo Bong Chae |
Imprint | New York, NY : Springer New York : Imprint: Springer, 1995 |
Edition | Second Edition |
Connect to | http://dx.doi.org/10.1007/978-1-4612-0781-8 |
Descript | XIII, 284 p. online resource |
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 Dirichletโs 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 TheoremโMonotone Convergence Theorem -- 6. The Lebesgue TheoremโDominated 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. Egoroffโs Theorem -- 8. Steinhausโ 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