本书共包含五章内容，主要介绍了分析学的关键——积分。第一章阐述了积分在分析学中的重要作用，详细地论述了常用的积分理论，给出了积分的相关概念；第二章和第三章讨论了黎曼积分和勒贝格积分；第四章对两种理论进行了比较；最后一章介绍了Henstock积分、Daniell积分、Riemann-Stieltjeg积分等常用积分。通过论述常用的积分理论及应用等知识，让读者更深入地了解有关分析学在积分中的重要性。本书由浅入深，详略得当，条理清晰，可供高等院校师生及数学爱好者参考使用。

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
编辑手记