Book:G. de Barra/Measure Theory and Integration

Subject Matter

 * Measure Theory

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



Source work progress
* : Chapter $1$: Preliminaries: $1.1$ Set Theory