測度論

本書是作者在莫斯科國立大學數學力學系的講稿基礎上編寫而成的：第一卷包括了通常測度論教材中的內容：測度的構造與延拓，Lebesgue積分的定義及基本性質，Jordan分解，Radon-Nikodym定理，Fourier變換，卷積，Lp空間，測度空間，Newton-Leibniz公式，極大函數，Henstock-Kurzweil；積分等。每章最後都附有非常豐富的補充與習題，其中包含許多有用的知識，例如：Whitney分解，Lebesgue-Stieltjes積分，Hausdorff測度，Brunn-Minkowski不等式，Hellinger積分與Heltinger距離，BMO類，Calderon-Zygmund分解等。書的最後有詳盡的參考文獻及歷史註記。這是一本很好的研究生教材和教學參考書。

Preface

Chapter 1. Constructions and extensions 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 extension 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 operations (48). Compact classes (50). Metric Boolean algebra (53).Measurable envelope, measurable kernel and inner measure (56).Extensions 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 extensions of Lebesgue measure (80). Whitney's decomposition (82). Exercises (83).

Chapter 2. The Lebesgue integral

2.1. Measurable functions

2.2. Convergence in measure and almost everywhere

2.3. The integral for simple functions

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. Connections with the Riemann integral

2.11. The HSlder and Minkowski inequalities

2.12. Supplements and exercises

The a-algebra generated by a class of functions (143). Borel mappings on IRn (145). The functional monotone class theorem (146). Baire classes of functions (148). Mean value theorems (150). The Lebesgue-Stieltjes integral (152). Integral inequalities (153). Exercises (156).

Chapter 3. Operations on measures and functions

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 transform

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). Decompositions of set functions (218). Properties of positive definite functions (220).The Brunn-Minkowski inequality and its generalizations (222).Mixed volumes (226). The Radon transform (227). Exercises (228).

Chapter 4. The spaces Lp and spaces of measures

4.1. The spaces Lp

4.2. Approximations 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 approximations in Lp (294).Certain conditions of convergence in Lp (298). Hellinger's integral and ellinger's distance (299). Additive set functions (302). Exercises (303).

Chapter 5. Connections between the integral and derivative.

5.1. Differentiability of functions on the real line

5.2. Functions of bounded variation

5.3. Absolutely continuous functions

5.4. The Newton-Leibniz formula

5.5. Covering theorems

5.6. The maximal function

5.7. The Henstock-Kurzweil integral

5.8. Supplements and exercises

Covering theorems (361). Density 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