Preface
About This Book
Acknowledgments
Contents
About the Author
1 Measure Theory
1.1 Notion of Measure
1.1.1 Motivation
1.1.2 Extended Real Numbers
1.1.3 Axiom of Measure
1.2 Outer Measure of a Set
1.2.1 Definition of Outer Measure
1.2.2 Basic Properties of Outer Measure
1.2.3 Additive Property of Outer Measure
1.3 Inner Measure
1.3.1 Definition of Inner Measure
1.3.2 Properties of Inner Measure
1.3.3 The Inner Measure Problem
1.3.4 Inner Measure As a Supremum
1.4 Lebesgue Measurability
1.4.1 Measurable Sets
1.4.2 Characterization of Measurable Sets
1.4.3 Measurable Sets as Sigma-Algebra
1.4.4 Borel Sets
1.4.5 Additivity of Lebesgue Measure
1.4.6 Continuity of Lebesgue Measure
1.5 Caratheodory Criterion
1.5.1 The Idea of Caratheodory's Approach
1.5.2 Characterization of Measurable Sets
1.5.3 Lebesgue Versus Caratheodory Approaches
1.5.4 Sigma-Algebra of Measurable Sets By Caratheodory Criterion
1.5.5 Additivity
1.6 Unusual Sets
1.6.1 Vitali Sets
1.6.2 Nonmeasurable Sets
1.6.3 Axiomatic Controversy
1.6.4 Cantor Set
1.7 Lebesgue Product Measure
1.7.1 The Measure of Rectangles and Cubes
1.7.2 Caratheodory Criterion in mathbbRn
1.7.3 Measure of Product Sets
1.8 Problems
2 Measurable Functions
2.1 Introduction
2.2 Measurable Functions
2.2.1 Simple Function
2.2.2 Simple Versus Step Functions
2.2.3 Definition of Measurable Function
2.2.4 Algebra of Measurable Functions
2.2.5 Almost Everywhere Property
2.3 Sequence of Measurable Functions
2.3.1 Supremum and Infimum
2.3.2 Limits of Sequences
2.3.3 Modes of Convergence
2.4 Approximation Theorems
2.4.1 Nearly Versus A.E.
2.4.2 First Littlewood Principle
2.4.3 Approximation of Simple Functions
2.4.4 Simple Approximation Theorem
2.4.5 Egoroff Theorem
2.4.6 Lusin Theorem
2.5 Differentiability
2.5.1 Dini Derivative
2.5.2 Vitali Covering
2.5.3 Derivative of Monotone Functions
2.6 Functions of Bounded Variation
2.6.1 Total Variation
2.6.2 Bounded Variation
2.6.3 Jordan Decomposition
2.7 Absolute Continuity
2.7.1 Absolute Continuous Functions
2.7.2 Functions of Bounded Derivatives
2.7.3 Derivative of Absolute Continuous Function
2.8 Problems
3 Lebesgue Integration
3.1 The Riemann Integral
3.1.1 Introduction
3.1.2 Riemann Integral of a Bounded Function
3.1.3 Deficiencies of Riemann Integral
3.2 Integral of Bounded Measurable Functions
3.2.1 The Idea of Lebesgue Integral
3.2.2 Integral of Simple Functions
3.2.3 Integral of Bounded Functions
3.3 Integral of General Measurable Functions
3.3.1 The Indeterminate Value Problem
3.3.2 Integral of Nonnegative Functions
3.3.3 The General Lebesgue Integral
3.4 Convergence Theorems
3.4.1 Uniform Convergence
3.4.2 Bounded Convergence Theorem
3.4.3 Monotone Convergence Theorem
3.4.4 Fatou's Lemma
3.4.5 Dominated Convergence Theorem
3.4.6 Historical Remark
3.5 Lebesgue Integrability
3.5.1 Riemann Integrability
3.5.2 Improper Riemann Integral
3.5.3 Lebesgue Integrability
3.5.4 Improper Riemann Integral Versus Lebesgue Integral
3.5.5 Riemann–Lebesgue Lemma
3.6 Lebesgue Fundamental Theorem of Calculus
3.6.1 The Recovering Problem
3.6.2 Integrability of The Derivative
3.6.3 Differentiation of Integral
3.6.4 Indefinite Integral of Derivative
3.7 Lebesgue Double Integral
3.7.1 The Double Integral
3.7.2 Sections of Sets and Functions
3.7.3 Fubini–Tonelli Theorems
3.7.4 Convolution
3.8 Problems
4 Lebesgue Spaces
4.1 Norms and Linear Spaces
4.1.1 Finite-Dimensional Linear Spaces
4.1.2 Definition of Norm
4.1.3 The p-Norm
4.2 Basic Theory of Lp Spaces
4.2.1 The Space of Lebesgue Integrable Functions
4.2.2 Definition of Lp Space
4.2.3 Linfty Spaces
4.2.4 Inclusions of Lp Spaces
4.3 Fundamental Inequalities
4.3.1 Young's Inequality
4.3.2 Holder Inequality
4.3.3 Minkowski Inequality
4.4 Further Spaces of Order p
4.4.1 Lp Spaces, 0
4.4.2 ellp Spaces
4.4.3 mathcalRp Spaces
4.5 Convergence in Lp
4.5.1 Convergence in p-Norm
4.5.2 Comparison Between Types of Convergence
4.5.3 p-Norm Dominated Convergence Theorem
4.5.4 Inclusion Relations
4.5.5 Convergence in Linfty
4.6 Approximations in Lp
4.6.1 Dense Subspaces
4.6.2 Density Results in Lp
4.6.3 Density Results in Linfty
4.7 Bounded Linear Functionals on Lp
4.7.1 Notion of Functional
4.7.2 Integral Functional on Lp
4.7.3 Riesz Representation Theorem
4.7.4 The Case p=infty
4.8 Problems
5 Abstract Measure Theory
5.1 Generalization of Measure Theory
5.1.1 Introduction
5.1.2 Measurable and Measure Spaces
5.1.3 Measurable Functions
5.1.4 Integration Over Abstract Measurable Spaces
5.2 Signed Measure
5.2.1 Notion of Signed Measure
5.2.2 Positive and Negative Sets
5.2.3 Hahn's Lemma
5.3 Decomposition of Measures
5.3.1 Hahn Decomposition Theorem
5.3.2 Jordan Decomposition Theorem
5.4 Absolute Continuity of Measures
5.4.1 Motivating Problem
5.4.2 Definitions
5.5 Radon–Nikodym Theorem
5.5.1 Radon–Nikodym Theorem for Finite Measures
5.5.2 Extended Radon-Nikodym Theorem
5.6 Radon–Nikodym Derivative
5.6.1 Fundamental Theorem of Calculus for RN Derivatives
5.6.2 Calculus of RN Derivatives
5.6.3 Chain Rule of RN Derivative
5.7 Lebesgue Decomposition of Measures
5.7.1 Mutually Singular Measures
5.7.2 Lebesgue Decomposition Theorem
5.8 Problems
Appendix Answer Key
Bibliography
Index
