Measure and Integral Theory and Practice

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 σ-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 σ-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 L^p(X, µ)

10.3   Completeness of L^p(X, µ)

10.4   Dense Sets in L^p(X, µ)

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 L^p(X, µ)

12.4   The Dual Space of L^∞(X, µ)

12.5   The Dual Space of C(X)

12.6   Weak Convergence

Epilogue                                                                                                                                                                                            

Solutions and Answers to Selected Exercises                                                                                                           

Bibliography

Subject Index                                                                                                                                                                                 

Author Index 

Biography

John Srdjan Petrovic was born in Belgrade, Yugoslavia. He earned his PhD from the University of Michigan under the direction of Dr.Carl Pearcy. His research area is the theory of operators on Hilbert space, and he has published more than 30 articles in prestigious journals. He is a professor of mathematics at Western Michigan University and his visiting positions include Texas A&M University, Indiana University, and University of North Carolina Charlotte. His text, Advanced Caluclus: Theory and Practice, is in its second edition (CRC Press).