Measure theory and integration 2nd Edition by G De Barra ISBN 1904275044 978-1904275046
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Product details:
ISBN 10: 1904275044
ISBN 13: 978-1904275046
Author: G De Barra
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 which test comprehension and for which detailed solutions are provided.
Table of contents:
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
Chapter 3 Integration of Functions of a Real Variable
3.1 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
Chapter 4 Differentiation
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
5.6 Integration with respect to a Measure
Chapter 6 Inequalities and the LpL^pLp Spaces
6.1 The LpL^pLp Spaces
6.2 Convex Functions
6.3 Jensen’s Inequality
6.4 The Inequalities of Hölder and Minkowski
6.5 Completeness of Lp(μ)L^p(mu)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
8.5 Bounded Linear Functionals on LpL^pLp
Chapter 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)C(I)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
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G De Barra,Measure theory