This book is prepared with two purpose: one is for a course of university level on abstract measure, integration theory and functional analysis, and another one is for a self-contained reference for whom interested in real analysis. This book is made up with clear explanations and elementary proofs to study the measure, integration theory and functional alone easily. However, the required prerequisite (pre-study) to read this book is the first course in analysis and a foundation of topology.

The measure and integration theory becomes more important as one of most fundamental theories in the present century for wide areas of applications including  mathematical physics, finance, biology and etc. There are several text books for the measure and integration theory with two different approaches. One is starting with the Lebesgue measure and integration theory to understand several motivations and examples, and then it is extended to abstract measure and integration theory. Another is starting abstract measure and integration theory and then the Lebesgue measure and integration theory is dealt and understood as an example. Both approaches have their own advantages and disadvantages. For this book, I have chosen the latter and so this book is starting with abstract measure theory and then dealing the Lebesgue measure and integration theory as an example, and so I hope that the readers can reach quickly to the measure and integration theory and then can apply it depending on their own interests. 

Chapter 1 offers fundamental notions in the first course in analysis and topology, which are necessary to read the main parts of this book. Chapters 2 and 3 deals with the general measure theory including constructions of abstract measures and the Lebesgue measure as an example of abstract measure. The readers interesting to the integration theory can study only the definition of abstract measure with fundamental properties and then can skip the constructions of abstract measures and the Lebesgue measure. Chapter 4 deals with measurable functions which is necessary for the abstract integration theory Chapter 5 provides the general theory of abstract integration and considers the Lebesgue integral as an example of an abstract integral The Lebeasgue integral is considered as an extension of the Riemann integral. Chapter 6 offers the Radon-Nikodym theorem, the fundamental theorem in calculus and Riemann-Stieltjes integral. Chapters 7 and 8 provides the fundamental notions in functional analysis. Chapter 9 considers the convergence of measurable functions including pointwise convergence, convergence in measure, almost uniform convergence and convergence in mean, among them the convergence in measure is weakest. Chapter 10 provides the fundamental theorems in functional analysis. Exercises are given at the end of each chapter and solutions to some problems are given. 

Preface iii
1 Preliminaries 1
1.1 Partially Ordered Set . 1
1.2 Completeness Axiom and Archimedian Property . 2
1.3 Countable and Uncountable Sets . 4
1.4 Open and Closed Sets  . 7
1.5 Sequences  . 13
1.6 Continuous Functions  . 20
1.7 Urysohn’s Lemma and Tietze Extension Theorem  . 27
 

2 General Measure Theory 35
2.1 Extended Real Number System  . 35
2.2 Measurable Spaces  . 35
2.3 Measure Spaces  . 39
2.4 SignedMeasure Spaces  . 46
2.5 OuterMeasures  . 53
 

3 Lebesgue Measurable Sets and Lebesgue Measure 71
3.1 Lebesgue OuterMeasure . 72
3.2 LebesgueMeasurable Sets  . 77
3.3 LebesgueMeasure  . 84
3.4 Nonmeasurable Sets  . 86
3.5 Cantor Set . 88
3.6 Cantor-Lebesgue Function  . 90
 

4 Measurable Functions 95
4.1 Measurable Functions  . 95
4.2 Littlewood’s Three Principles  . 108
 

5 General Integration Theory 113
5.1 Integration over GeneralMeasure Spaces  . 113
5.2 Lebesgue Integration  . 130
5.3 Riemann Integral as Lebsgue Integral . 132
5.4 Lebesgue’s Theorem for Riemann Integrability . 134
 

6 Differentiation and Integration 141
6.1 Uniform Integrability and Tightness . 142
6.2 Radon-NikodymTheorem  . 147
6.3 Functions of Bounded Variation  . 152
6.4 Differentiability of Functions of Bounded Variation . 155
6.5 Differentiation of Integral  . 164
6.6 Absolutely Continuous Functions  . 168
6.7 Riemann-Stieltjes Integral . 174
6.8 Convex Functions . 180
 

7 The Lp(X) Spaces 187
7.1 Seminormed and Normed Spaces  . 187
7.2 Bounded Linear Operators on Normed Spaces . 200
7.3 Lp Spaces . 209
7.4 Inequalities . 211
7.5 Completeness of Lp(X)  . 218
7.6 Dual of Lp(X) . 223
7.7 Separability of Lp  . 229
 

8 Product Measures and Integration 237
8.1 ProductMeasures  . 237
8.2 Integration on ProductMeasure Spaces  . 240
 

9 Convergence of Measurable Functions 249
9.1 Convergence inMeasure . 249
9.2 Almost UniformConvergence  . 253
9.3 Convergence inMean  . 255
 

10 Further Topics in Functional Analysis 259
10.1 Hahn-Banach Theorem  . 261
10.2 Baire Category Theorem  . 265
10.3 Uniform Boundedness Principle  . 269
10.4 Open Mapping Theorem  . 270
10.5 Closed Graph Theorem  . 272
10.6 Weak Topology . 273
 

Index 313

Un Cig Ji
Chungbuk National University Department of Mathematics Professor

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