Preface 1 Introduction 1.1 What is an Integral? 1.2 What is the Riemann Integral? 1.3 What is the Riemann Integral Good For? 1.4 What is the Riemann Integral Not Good For? 1.5 What is the Lebesgue Integral? 1.6 What is the Lebesgue Integral Good For? 1.7 What is the Lebesgue Not Good For? Exercises 2 The Riemann Integral 2.1 The Definition 2.2 Properties of the Riemann Integral 2.3 Characterization of Riemann Integrability 2.4 The Fundamental Theorem of Calculus 2.5 NumericatTechniques of Integration 2.5.1 Introduction 2.5.2 The Method of Rectangles 2.5.3 The Trapezoidal Rule 2.5.4 Simpson's Rule 2.6 Integration by Parts Exercises 3 The Lebesgue Integral 3.1 Elementary Measure Theory 3.2 Measurable Sets 3.3 The Lebesgue Integral 3.4 Three Big Theorems about the Lebesgue Integral 3.5 The Lebesgue Spaces LP 3.6 The Riesz Representation Theorem 3.7 Product Integration: Fubini's Theorem 3.8 Three Principles of Littlewood 3.9 Differentiation of Integrals: Covering Lemmas and the Lebesgue Theorem 3.9.1 Basic Ideas 3.9.2 The Maximal Function 3.10 The Concept of Convergence in Measure 3.11 Functions of Bounded Variation and Absolute Continuity Exercises 4 Comparison of the Riemann and Lebesgue Integrals 4.1 Any Riemann Integrable Function is Lebesgue Integrable Exercises 5 Other Theories of the Integral 5.1 The Daniell Integral 5.2 The Riemann-Stieltjeg Integral 5.3 The Henstock-Kurzweil Integral 5.4 Hausdorff Measure 5.5 Haar Measure 5.5.1 The Fundamental Theorem Exercises Bibliography Author's Biography Index ��݋��ӛ