Measure and Integral

Prologue

I Preliminaries

1 Set Theory

1.1 Sets

1.2 Functions

1.3 Cardinal and Ordinal Numbers

1.4 The Axiom of Choice

2 Metric Spaces

2.1 Elementary Theory of Metric Spaces

2.2 Completeness

2.3 Compactness

2.4 Limits of Functions

2.5 Baire's Theorem

3 Geometry of the Line and the Plane

II Measure Theory

4 Lebesgue Measure on R2

4.1 Jordan Measure

4.2 Lebesgue Measure

4.3 The ?-Algebra of Lebesgue Measurable Sets

5 Abstract Measure

5.1 Measures and Measurable Sets

5.2 Carath?eodory Extension of Measure

5.3 Lebesgue Measure on Euclidean Spaces

5.4 Beyond Lebesgue ?-Algebra

5.5 Signed Measures

6 Measurable Functions

6.1 Definition and Basic Facts

6.2 Fundamental Properties of Measurable Functions

6.3 Sequences of Measurable Functions

III Integration Theory

7 The Integral

7.1 About Riemann Integral

7.2 Integration of Nonnegative Measurable Functions

7.3 The Integral of a Real-Valued Function

7.4 Computing Lebesgue Integral

8 Integration on Product Spaces

8.1 Measurability on Cartesian Products

8.2 Product Measures

8.3 The Fubini Theorem

9 Differentiation and Integration

9.1 Dini Derivatives

9.2 Monotone Functions

9.3 Functions of Bounded Variation

9.4 Absolutely Continuous Functions

9.5 The Radon-Nikodym Theorem

IV An Introduction to Functional Analysis

10 Banach Spaces

10.1 Normed Linear Spaces

10.2 The Space Lp(X, ?)

10.3 Completeness of Lp(X, ?)

10.4 Dense Sets in Lp(X, ?)

10.5 Hilbert Space

10.6 Bessel's Inequality and Orthonormal Bases

10.7 The Space C(X)

11 Continuous Linear Operators Between Banach Spaces

11.1 Linear Operators

11.2 Banach Space Isomorphisms

11.3 The Uniform Boundedness Principle

11.4 The Open Mapping and Closed Graph Theorems

12 Duality

12.1 Linear Functionals

12.2 The Hahn-Banach Theorem

12.3 The Dual of Lp(X, ?)

12.4 The Dual Space of L?(X, ?)

12.5 The Dual Space of C(X)

12.6 Weak Convergence

Epilogue

Solutions and Answers to Selected Exercises

Bibliography

Subject Index

Author Index