Real Analysis

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