测度论（第一卷）（影印版） / 天元基金影印数学丛书
￥39.50定价
作者： V.I.Bogachev
出版时间：2010年7月
出版社：高等教育出版社
- 高等教育出版社
- 9787040286960
- 1版
- 125153
- 0045150269-4
- 异16开
- 2010年7月
- 450
- 500
- 理学
- 数学
- O174.12
- 数学类
- 研究生、本科
内容简介
本书是作者在莫斯科国立大学数学力学系的讲稿基础上编写而成的：第一卷包括了通常测度论教材中的内容：测度的构造与延拓，Lebesgue积分的定义及基本性质，Jordan分解，Radon-Nikodym定理，Fourier变换，卷积，Lp空间，测度空间，Newton-Leibniz公式，极大函数，Henstock-Kurzweil；积分等。每章最后都附有非常丰富的补充与习题，其中包含许多有用的知识，例如：Whitney分解，Lebesgue-Stieltjes积分，Hausdorff测度，Brunn-Minkowski不等式，Hellinger积分与Heltinger距离，BMO类，Calderon-Zygmund分解等。书的最后有详尽的参考文献及历史注记。这是一本很好的研究生教材和教学参考书。
目录
Preface
Chapter 1 Cotructio and exteio of measures
1.1 Measurement of length: introductory remarks
1.2 Algebras and σ-algebras
1.3 Additivity and countable additivity of measures
1.4 Compact classes and countable additivity
1.5 Outer measure and the Lebesgue exteion of measures
1.6 Infinite and a-finite measures
1.7 Lebesgue measure
1.8 Lebesgue-Stieltjes measures
1.9 Monotone and σ-additive classes of sets
1.10 Souslin sets and the A-operation
1.11 Caratheodory outer measures
1.12 Supplements and exercises
Set operatio (48) Compact classes (50) Metric Boolean algebra
(53).Measurable envelope, measurable kernel and inner measure
(56).Exteio of measures (58) Some interesting sets (61) Additive,
but not countably additive measures (67) Abstract inner measures
(70).Measures on lattices of sets (75) Set-theoretic problems in
measure theory (77) Invariant exteio of Lebesgue measure (80)
Whitney's decomposition (82) Exercises (83)
Chapter 2 The Lebesgue integral
2.1 Measurable functio
2.2 Convergence in measure and almost everywhere
2.3 The integral for simple functio
2.4 The general definition of the Lebesgue integral
2.5 Basic properties of the integral
2.6 Integration with respect to infinite measures
2.7 The completeness of the space L1
2.8 Convergence theorems
2.9 Criteria of integrability
2.10 Connectio with the Riemann integral
2.11 The HSlder and Minkowski inequalities
2.12 Supplements and exercises
The a-algebra generated by a class of functio (143) Borel
mappings on IRn (145) The functional monotone class theorem (146)
Baire classes of functio (148) Mean value theorems (150) The
Lebesgue-Stieltjes integral (152) Integral inequalities (153)
Exercises (156)
Chapter 3 Operatio on measures and functio
3.1 Decomposition of signed measures
3.2 The Radon-Nikodym theorem
3.3 Products of measure spaces
3.4 Fubini's theorem
3.5 Infinite products of measures
3.6 Images of measures under mappings
3.7 Change of variables in IRn
3.8 The Fourier traform
3.9 Convolution
3.10 Supplements and exercises
On Fubini's theorem and products of σ-algebras (209) Steiner's
symmetrization (212) Hausdorff measures (215) Decompositio of set
functio (218) Properties of positive definite functio (220).The
Brunn-Minkowski inequality and its generalizatio (222).Mixed
volumes (226) The Radon traform (227) Exercises (228)
Chapter 4 The spaces Lp and spaces of measures
4.1 The spaces Lp
4.2 Approximatio in Lp
4.3 The Hilbert space L2
4.4 Duality of the spaces Lp
4.5 Uniform integrability
4.6 Convergence of measures
4.7 Supplements and exercises
The spaces Lp and the space of measures as structures (277)
The weak topology in LP(280) Uniform convexity of LP(283) Uniform
integrability and weak compactness in L1 (285) The topology of
setwise convergence of measures (291) Norm compactness and
approximatio in Lp (294).Certain conditio of convergence in Lp
(298) Hellinger's integral and ellinger's distance (299) Additive
set functio (302) Exercises (303)
Chapter 5 Connectio between the integral and derivative
5.1 Differentiability of functio on the real line
5.2 Functio of bounded variation
5.3 Absolutely continuous functio
5.4 The Newton-Leibniz formula
5.5 Covering theorems
5.6 The maximal function
5.7 The Hetock-Kurzweil integral
5.8 Supplements and exercises
Covering theorems (361) Deity points and Lebesgue points
(366).Differentiation of measures on IRn (367) The approximate
continuity (369) Derivates and the approximate differentiability
(370).The class BMO (373) Weighted inequalities (374) Measures
with the doubling property (375) Sobolev derivatives (376) The
area and coarea formulas and change of variables (379) Surface
measures (383).The Calder6n-Zygmund decomposition (385) Exercises
(386)
Bibliographical and Historical Comments
References
Author Index
Subject Index
Chapter 1 Cotructio and exteio of measures
1.1 Measurement of length: introductory remarks
1.2 Algebras and σ-algebras
1.3 Additivity and countable additivity of measures
1.4 Compact classes and countable additivity
1.5 Outer measure and the Lebesgue exteion of measures
1.6 Infinite and a-finite measures
1.7 Lebesgue measure
1.8 Lebesgue-Stieltjes measures
1.9 Monotone and σ-additive classes of sets
1.10 Souslin sets and the A-operation
1.11 Caratheodory outer measures
1.12 Supplements and exercises
Set operatio (48) Compact classes (50) Metric Boolean algebra
(53).Measurable envelope, measurable kernel and inner measure
(56).Exteio of measures (58) Some interesting sets (61) Additive,
but not countably additive measures (67) Abstract inner measures
(70).Measures on lattices of sets (75) Set-theoretic problems in
measure theory (77) Invariant exteio of Lebesgue measure (80)
Whitney's decomposition (82) Exercises (83)
Chapter 2 The Lebesgue integral
2.1 Measurable functio
2.2 Convergence in measure and almost everywhere
2.3 The integral for simple functio
2.4 The general definition of the Lebesgue integral
2.5 Basic properties of the integral
2.6 Integration with respect to infinite measures
2.7 The completeness of the space L1
2.8 Convergence theorems
2.9 Criteria of integrability
2.10 Connectio with the Riemann integral
2.11 The HSlder and Minkowski inequalities
2.12 Supplements and exercises
The a-algebra generated by a class of functio (143) Borel
mappings on IRn (145) The functional monotone class theorem (146)
Baire classes of functio (148) Mean value theorems (150) The
Lebesgue-Stieltjes integral (152) Integral inequalities (153)
Exercises (156)
Chapter 3 Operatio on measures and functio
3.1 Decomposition of signed measures
3.2 The Radon-Nikodym theorem
3.3 Products of measure spaces
3.4 Fubini's theorem
3.5 Infinite products of measures
3.6 Images of measures under mappings
3.7 Change of variables in IRn
3.8 The Fourier traform
3.9 Convolution
3.10 Supplements and exercises
On Fubini's theorem and products of σ-algebras (209) Steiner's
symmetrization (212) Hausdorff measures (215) Decompositio of set
functio (218) Properties of positive definite functio (220).The
Brunn-Minkowski inequality and its generalizatio (222).Mixed
volumes (226) The Radon traform (227) Exercises (228)
Chapter 4 The spaces Lp and spaces of measures
4.1 The spaces Lp
4.2 Approximatio in Lp
4.3 The Hilbert space L2
4.4 Duality of the spaces Lp
4.5 Uniform integrability
4.6 Convergence of measures
4.7 Supplements and exercises
The spaces Lp and the space of measures as structures (277)
The weak topology in LP(280) Uniform convexity of LP(283) Uniform
integrability and weak compactness in L1 (285) The topology of
setwise convergence of measures (291) Norm compactness and
approximatio in Lp (294).Certain conditio of convergence in Lp
(298) Hellinger's integral and ellinger's distance (299) Additive
set functio (302) Exercises (303)
Chapter 5 Connectio between the integral and derivative
5.1 Differentiability of functio on the real line
5.2 Functio of bounded variation
5.3 Absolutely continuous functio
5.4 The Newton-Leibniz formula
5.5 Covering theorems
5.6 The maximal function
5.7 The Hetock-Kurzweil integral
5.8 Supplements and exercises
Covering theorems (361) Deity points and Lebesgue points
(366).Differentiation of measures on IRn (367) The approximate
continuity (369) Derivates and the approximate differentiability
(370).The class BMO (373) Weighted inequalities (374) Measures
with the doubling property (375) Sobolev derivatives (376) The
area and coarea formulas and change of variables (379) Surface
measures (383).The Calder6n-Zygmund decomposition (385) Exercises
(386)
Bibliographical and Historical Comments
References
Author Index
Subject Index
