UID:
edoccha_9958072444802883
Format:
1 online resource (240 p.)
Edition:
2nd ed.
ISBN:
0-85709-952-3
Content:
This text approaches integration via measure theory as opposed to measure theory via integration, an approach which makes it easier to grasp the subject. Apart from its central importance to pure mathematics, the material is also relevant to applied mathematics and probability, with proof of the mathematics set out clearly and in considerable detail. Numerous worked examples necessary for teaching and learning at undergraduate level constitute a strong feature of the book, and after studying statements of results of the theorems, students should be able to attempt the 300 problem exercises whi
Note:
"First published 1981"--T.p. verso.
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Front Cover; ABOUT OUR AUTHOR; Measure Theory and Integration; Copyright Page; Table of Contents; Preface to First Edition; Notation; Chapter 1. Preliminaries; 1.1 Set Theory; 1.2 Topological Ideas; 1.3 Sequences and Limits; 1.4 Functions and Mappings; 1.5 Cardinal Numbers and Countability; 1.6 Further Properties of Open Sets; 1.7 Cantor-like Sets; Chapter 2. Measure on the Real Line; 2.1 Lebesgue Outer Measure; 2.2 Measurable Sets; 2.3 Regularity; 2.4 Measurable Functions; 2.5 Borel and Lebesgue Measurability; 2.6 Hausdorff Measures on the Real Line
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Chapter 3. Integration of Functions of a Real Variable3.1 Integration of Non-negative Functions; 3.2 The General Integral; 3.3 Integration of Series; 3.4 Riemann and Lebesgue Integrals; Chapter 4. Differentiation; 4.1 The Four Derivates; 4.2 Continuous Non-differentiable Functions; 4.3 Functions of Bounded Variation; 4.4 Lebesgue's Differentiation Theorem; 4.5 Differentiation and Integration; 4.6 The Lebesgue Set; Chapter 5. Abstract Measure Spaces; 5.1 Measures and Outer Measures; 5.2 Extension of a Measure; 5.3 Uniqueness of the Extension; 5.4 Completion of a Measure; 5.5 Measure Spaces
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5.6 Integration with respect to a MeasureChapter 6. Inequalities and the Lp Spaces; 6.1 The Lp Spaces; 6.2 Convex Functions; 6.3 Jensen's Inequality; 6.4 The Inequalities of Hölder and Minkowski; 6.5 Completeness of Lp(μ); Chapter 7. Convergence; 7.1 Convergence in Measure; 7.2 Almost Uniform Convergence; 7.3 Convergence Diagrams; 7.4 Counterexamples; Chapter 8. Signed Measures and their Derivatives; 8.1 Signed Measures and the Hahn Decomposition; 8.2 The Jordan Decomposition; 8.3 The Radon-Nikodym Theorem; 8.4 Some Applications of the Radon-Nikodym Theorem
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8.5 Bounded Linear Functional on LpChapter 9. Lebesgue-Stieltjes Integration; 9.1 Lebesgue-Stieltjes Measure; 9.2 Applications to Hausdorff Measures; 9.3 Absolutely Continuous Functions; 9.4 Integration by Parts; 9.5 Change of Variable; 9.6 Riesz Representation Theorem for C(I); Chapter 10. Measure and Integration in a Product Space; 10.1 Measurability in a Product Space; 10.2 The Product Measure and Fubini's Theorem; 10.3 Lebesgue Measure in Euclidean Space; 10.4 Laplace and Fourier Transforms; Hints and Answers to Exercises; Chapter 1; Chapter 2; Chapter 3; Chapter 4; Chapter 5; Chapter 6
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Chapter 7Chapter 8; Chapter 9; Chapter 10; References; Index
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English
Additional Edition:
ISBN 1-904275-04-4
Language:
English
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