This text contains a basic introduction to the abstract measure theory and the Lebesgue integral. Most of the standard topics in the measure and integration theory are discussed. In addition, topics on the Hewitt-Yosida decomposition, the Nikodym and Vitali-Hahn-Saks theorems and material on finitely additive set functions not contained in standard texts are explored. There is an introductory section on functional analysis, including the three basic principles, which is used to discuss many of the classic Banach spaces of functions and their duals. There is also a chapter on Hilbert space and the Fourier transform. TABLE OF CONTENTS......Page 7 Preface......Page 5 1.1 Preliminaries ......Page 11 1.2 Extended Real Numbers (R^*) and R^n ......Page 14 1.3 Lebesgue's Definition of the Integral ......Page 19 2.1 Semi-rings and Algebras of Sets ......Page 25 2.2 Additive Set Functions ......Page 30 2.2.1 Jordan Decomposition ......Page 38 2.2.2 Hahn Decomposition ......Page 43 2.2.3 Drewnowski's Lemma ......Page 45 2.3 Outer Measures ......Page 46 2.3.1 Metric Outer Measures ......Page 49 2.4 Extensions of Premeasures ......Page 51 2.4.1 Hewitt-Yosida Decomposition ......Page 59 2.5 Lebesgue Measure ......Page 60 2.6 Lebesgue-Stieltjes Measure ......Page 67 2.6.1 Hewitt-Yosida Decomposition for Lebesgue-Stieltjes Measures ......Page 70 2.7 Regular Measures ......Page 72 2.8 The Nikodym Convergence and Boundedness Theorems ......Page 76 3.1 Measurable Functions ......Page 81 3.1.1 Approximation of Measurable Functions ......Page 88 3.2 The Lebesgue Integral ......Page 91 3.3 The Riemann and Lebesgue Integrals ......Page 106 3.4 Integrals Depending on a Parameter ......Page 110 3.5 Convergence in Mean ......Page 113 3.6 Convergence in Measure ......Page 117 3.7 Comparison of Modes of Convergence ......Page 122 3.8 Mikusinski's Characterization of the Lebesgue Measure ......Page 125 3.9 Product Measures and Fubini's Theorem ......Page 128 3.10 A Geometric Interpretation of the Integral ......Page 136 3.11 Convolution Product ......Page 137 3.12 The Radon-Nikodym Theorem ......Page 142 3.12.1 The Radon-Nikodym Theorem for Finitely Additive Set Functions ......Page 147 3.13 Lebesgue Decomposition ......Page 151 3.14 The Vitali-Hahn-Sake Theorem ......Page 154 4.1 Differentiating Indefinite Integrals ......Page 157 4.2 Differentiation of Monotone Functions ......Page 162 4.3 Integrating Derivatives ......Page 168 4.4 Absolutely Continuous Functions ......Page 170 5.1 Normed Linear Spaces (NLS) ......Page 175 5.2 Linear Mappings between Normed Linear Spaces ......Page 184 5.3 The Uniform Boundedness Principle ......Page 188 5.4 Quotient Spaces ......Page 191 5.5 The Closed Graph/Open Mapping Theorems ......Page 193 5.6 The Hahn-Banach Theorem ......Page 196 5.6.1 Applications of the Hahn-Banach Theorem in NLS ......Page 199 5.6.2 Extension of Bounded, Finitely Additive Set Function s ......Page 203 5.6.3 A Translation Invariant, Finitely Additive Set Function ......Page 204 5.7 Ordered Linear Spaces ......Page 207 6.1 L^P-spaces, 1
