作者 Sterling K.Berberian 著
出版社 世界图书出版公司
出版时间 2004-04
版次 1
装帧 平装
货号 A23
上书时间 2022-08-06
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图书标准信息
作者
Sterling K.Berberian 著
出版社
世界图书出版公司
出版时间
2004-04
版次
1
ISBN
9787506266161
定价
63.00元
装帧
平装
开本
其他
纸张
胶版纸
页数
479页
【内容简介】
This book is a record of a course on functions of a real variable, addressed to first-year graduate students in mathematics, offered in the academic year 1985-86 at the University of Texas at Austin. It consists essentially of the day-by-day lecture notes that I prepared for the course, padded up with the exercises that I seemed never to have the time to prepare in advance; the structure and contents of the course are preserved faithfully, with minor cosmetic changes here and there.
【目录】
Preface CHAPTER 1 Foundations 1.1. Logic, set notations 1.2. Relations 1.3. Functions (mappings) 1.4. Product sets, axiom of choice 1.5. Inverse functions 1.6. Equivalence relations, partitions, quotient sets 1.7. Order relations 1.8. Real numbers . 1.9. Finite and infinite sets 1.10. Countable and uncountable sets 1.1 1. Zorn's lemma, the well-ordering theorem 1.12. Cardinality 1.13. Cardinal arithmetic, the continuum hypothesis 1.14. Ordinality 1.15. Extended real numbers 1.16. limsup, liminf, convergence in CHAPTER 2 Lebesgue Measure 2.1. Lebesgue outer measure on 2.2. Measurable sets 2.3. Cantor set: an uncountable set of measure zero 2.4. Borel sets, regularity 2.5. A nonmeasurable set 2.6. Abstract measure spaces CHAPTER 3 Topology 3.1. Metric spaces: examples 3.2. Convergence, closed sets and open sets in metric spaces 3.3. Topological spaces 3.4. Continuity 3.5. Limit of a function CHAPTER 4 Lebesgue Integral 4.1. Measurable functions 4.2. a.e. 4.3. Integrable simple functions 4.4. Integrable functions 4.5. Monotone convergence theorem, Fatou's lemma 4.6. Monotone classes 4.7. Indefinite integrals 4.8. Finite signed measures CHAPTER. 5 Differentiation 5.1. Bounded variation, absolute continuity 5.2. Lebesgue's representation of AC functions 5.3. limsup, liminf of functions; Dini derivates 5.4. Criteria for monotonicity 5.5. Semicontinuity 5.6. Semicontinuous approximations of integrable functions 5.7. F. Riesz's "Rising sun lemma" 5.8. Growth estimates of a continuous increasing function 5.9. Indefinite integrals are a.e. primitives 5.10. Lebesgue's "Fundamental theorem of calculus" 5.11. Measurability of derivates of a monotone function 5.12. Lebesgue decomposition of a function of bounded variation 5.13. Lebesgue's criterion for Riemann-integrability CHAPTER 6 Function Spaces 6.1. Compact metric spaces 6.2. Uniform convergence, iterated limits theorem 6.3. Complete metric spaces 6.4. LI 6.5. Real and complex measures 6.6. Loo 6.7. Lp (1
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