Contents

Preface xi

Chapter 1. The Regulated and Riemann Integrals 1

§1.1. Introduction 1

§1.2. Basic Properties of an Integral 2

§1.3. Step Functions 4

§1.4. Uniform and Pointwise Convergence 7

§1.5. Regulated Integral 8

§1.6. The Fundamental Theorem of Calculus 13

§1.7. The Riemann Integral 16

Chapter 2. Lebesgue Measure 25

§2.1. Introduction 25

§2.2. Null Sets 27

§2.3. Sigma Algebras 29

§2.4. Lebesgue Measure 31

§2.5. The Lebesgue Density Theorem 35

§2.6. Lebesgue Measurable Sets – Summary 37

Chapter 3. The Lebesgue Integral 41

§3.1. Measurable Functions 41

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