Book:G. de Barra/Measure Theory and Integration

From ProofWiki
Jump to: navigation, search

G. de Barra: Measure Theory and Integration

Published $1981$, Ellis Horwood Ltd..


Subject Matter


Contents

Preface
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
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
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 $L^p$ Spaces
6.1 The $L^p$ Spaces
6.2 Convex Functions
6.3 Jensen's Inequality
6.4 The Inequalities of Holder and Minkowski
6.5 Completeness of $L^p \left({\mu}\right)$
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 $L^p$
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 $C \left({I}\right)$
Chapter 10 Measure and Integration In a Product Space
10.l 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
Chapter 7
Chapter 8
Chapter 9
Chapter 10
References
Index