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广义Riemann积分（影印版）

样章
作者：
Robert M. McLeod 著
定价：
135.00 元
版面字数：
480.00千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2026年
ISBN：
978-7-04-065972-6
物料号：
65972-00
出版时间：
2026-03-27
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
分析
暂无
  • 前辅文
  • INTRODUCTION
  • CHAPTER 1—DEFINITION OF THE GENERALIZED RIEMANN INTEGRAL
    • 1.1 Selecting Riemann sums
    • 1.2 Definition of the generalized Riemann integral
    • 1.3 Integration over unbounded intervals
    • 1.4 The fundamental theorem of calculus
    • 1.5 The status of improper integrals
    • 1.6 Multiple integrals
    • 1.7 Sum of a series viewed as an integral
    • S1.8 The limit based on gauges
    • S1.9 Proof of the fundamental theorem
    • 1.10 Exercises
  • CHAPTER 2—BASIC PROPERTTIES OF THE INTEGRAL
    • 2.1 The integral as a function of the integrand
    • 2.2 The Cauchy criterion
    • 2.3 Integrability on subintervals
    • 2.4 The additivity of integrals
    • 2.5 Finite additivity of functions of intervals
    • 2.6 Continuity of integrals. Existence of primitives
    • 2.7 Change of variables in integrals on intervals in -R
    • S2.8 Limits of integrals over expanding intervals
    • 2.9 Exercises
  • CHAPTER 3—ABSOLUTE INTEGRABILITY AND CONVERGENCE THEOREMS
    • 3.1 Henstock's lemma
    • 3.2 Integrability of the absolute value of an integrable function
    • 3.3 Lattice operations on integrable functions
    • 3.4 Uniformly convergent sequences of functions
    • 3.5 The monotone convergence theorem
    • 3.6 The dominated convergence theorem
    • S3.7 Proof of Henstock's lemma
    • S3.8 Proof of the criterion for integrability of |f|
    • S3.9 Iterated limits
    • S3.10 Proof of the monotone and dominated convergence theorems
    • 3.11 Exercises
  • CHAPTER 4—INTEGRATION ON SUBSETS OF INTERVALS
    • 4.1 Null functions and null sets
    • 4.2 Convergence almost everywhere
    • 4.3 Integration over sets which are not intervals
    • 4.4 Integration of continuous functions on closed, bounded sets
    • 4.5 Integrals on sequences of sets
    • 4.6 Length, area, volume, and measure
    • 4.7 Exercises
  • CHAPTER 5—MEASURABLE FUNCTIONS
    • 5.1 Measurable functions
    • 5.2 Measurability and absolute integrability
    • 5.3 Operations on measurable functions
    • 5.4 Integrability of products
    • S5.5 Approximation by step functions
    • 5.6 Exercises.
  • CHAPTER 6—MULTIPLE AND ITERATED INTEGRALS
    • 6.1 Fubini's theorem
    • 6.2 Determining integrability from iterated integrals
    • S6.3 Compound divisions. Compatibility theorem
    • S6.4 Proof of Fubini's theorem
    • S6.5 Double series
    • 6.6 Exercises
  • CHAPTER 7—INTEGRALS OF STIELTJES TYPE
    • 7.1 Three versions of the Riemann-Stieltjes integral
    • 7.2 Basic properties of Riemann-Stieltjes integrals
    • 7.3 Limits, continuity, and differentiability of integrals
    • 7.4 Values of certain integrals
    • 7.5 Existence theorems for Riemann-Stieltjes integrals
    • 7.6 Integration by parts
    • 7.7 Integration of absolute values. Lattice operations
    • 7.8 Monotone and dominated convergence
    • 7.9 Change of variables
    • 7.10 Mean value theorems for integrals
    • S7.11 Sequences of integrators
    • S7.12 Line integrals
    • S7.13 Functions of bounded variation and regulated functions
    • S7.14 Proof of the absolute integrability theorem
    • 7.15 Exercises
  • CHAPTER 8—COMPARISON OF INTEGRALS
    • S8.1 Characterization of measurable sets
    • S8.2 Lebesgue measure and integral
    • S8.3 Characterization of absolute integrability using Riemann sums
    • 8.4 Suggestions for further study
  • REFERENCES
  • APPENDIX Solutions of In-text Exercises
  • INDEX

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