图书信息
图书目录
广义Riemann积分（影印版）
暂无简介
作者：
Robert M. McLeod 著
定价：
135.00 元
出版时间：
2026-03-27
ISBN：
978-7-04-065972-6
物料号：
65972-00
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
重点项目：
暂无
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：
2026-01-06
- 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
