# Book:G. de Barra/Measure Theory and Integration

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## G. de Barra:

## 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**