Book:G. de Barra/Measure Theory and Integration

G. de Barra: Measure Theory and Integration

Published $\text {1981}$, Ellis Horwood Ltd.

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

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