測度論（第一卷 影印版）

前言

為了更好地借鑒國外數(shù)學(xué)教育與研究的成功經(jīng)驗，促進(jìn)我國數(shù)學(xué)教育與研究事業(yè)的發(fā)展，提高高等學(xué)校數(shù)學(xué)教育教學(xué)質(zhì)量，本著“為我國熱愛數(shù)學(xué)的青年創(chuàng)造一個較好的學(xué)習(xí)數(shù)學(xué)的環(huán)境”這一宗旨，天元基金贊助出版“天元基金影印數(shù)學(xué)叢書”。該叢書主要包含國外反映近代數(shù)學(xué)發(fā)展的純數(shù)學(xué)與應(yīng)用數(shù)學(xué)方面的優(yōu)秀書籍，天元基金邀請國內(nèi)各個方向的知名數(shù)學(xué)家參與選題的工作，經(jīng)專家遴選、推薦，由高等教育出版社影印出版。為了提高我國數(shù)學(xué)研究生教學(xué)的水平，暫把選書的目標(biāo)確定在研究生教材上。當(dāng)然，有的書也可作為高年級本科生教材或參考書，有的書則介于研究生教材與專著之間。歡迎各方專家、讀者對本叢書的選題、印刷、銷售等工作提出批評和建議。

內(nèi)容概要

　　《測度論（第1卷）（影印版）》是作者在莫斯科國立大學(xué)數(shù)學(xué)力學(xué)系的講稿基礎(chǔ)上編寫而成的。第一卷包括了通常測度論教材中的內(nèi)容：測度的構(gòu)造與延拓，Lebesgue積分的定義及基本性質(zhì)，Jordan分解，Radon-Nikodym定理，F(xiàn)ourier變換，卷積，L空間，測度空間，Newton-Leibniz公式，極大函數(shù)，Henstock-Kurzweil積分等每章最后都附有非常豐富的補(bǔ)充與習(xí)題，其中包含許多有用的知識，例如：Whitney分解，Lebesgue-Stieltjes積分，Hausdorff度，Brunn-Minkowski不等式，Hellinger積分與Hellinger距離，BMO類，Calderon-Zygmund分解等。書的最后有詳盡的參考文獻(xiàn)及歷史注記。這是一本很好的研究生教材和教學(xué)參考書。

作者簡介

作者：（俄羅斯）博根切維（V.l.Bogachev）

書籍目錄

PrefaceChapter 1.Constructions and extensions of measures1.1.Measurement of length: introductory remarks1.2.Algebras and c-algebras1.3.Additivity and countable additivity of measures1.4.Compact classes and countable additivity1.5.Outer measure and the Lebesgue extension of measures1.6.Infinite and a-finite measures1.7. Lebesgue measure1.8.Lebesgue-Stieltjes measures1.9.Monotone and a-additive classes of sets1.10.Souslin sets and the A-operation1.11.Carath~odory outer measures1.12. Supplements and exercisesSet 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 integral2.1.Measurable functions2.2.Convergence in measure and almost everywhere2.3. The integral for simple functions2.4.The general definition of the Lebesgue integral2.5.Basic properties of the integral2.6.Integration with respect to infinite measures2.7.The completeness of the space L12.8.Convergence theorems2.9.Criteria of integrability2.10.Connections with the Riemann integral2.11.The HSlder and Minkowski inequalities2.12.Supplements and exercisesThe a-Mgebra generated by a class of functions (143).The functional monotone class theorem (146).Balre classes of functions (148).Mean value theorems (150).The LebesgueStieltjes integral (152).Integral inequalities (153).Exercises (156).Chapter 3. Operations on measures and functions3.1.Decomposition of signed measures3.2.The Radon-Nikodym theorem3.3.Products of measure spaces3.4.F~abini's theorem3.5.Infinite products of measures3.6. Images of measures under mappings3.7.Change of variables in 3.8.The Fourier transform3.9.Convolution3.10. Supplements and exercisesOn Fubini's theorem and products of a-algebras (209).Steiner's symmetrization (212).Hausdorff measures (215).Decompositions ofset 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 measures4.1.The spaces Lp4.2.Approximations in Lp4.3.The Hilbert space L24.4.Duality of the spaces Lp4.5.Uniform integrability 4.6.Convergence of measures4.7.Supplements and exercisesThe spaces Lp and the space of measures as structures (277).The weaktopology in LP(280).Uniform convexity of LP(283).Uniform integrabilityand 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 andHellinger'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 line5.2.Functions of bounded variation5.3.Absolutely continuous functions5.4.The Newton-Leibniz formula5.5.Covering theorems5.6.The maximal function5.7.The Henstock-Kurzweil integral5.8.Supplements and exercisesCovering theorems (361).Density points and Lebesgue points (366).Differentiation of measures on ]Rn (367). The approximatecontinuity (369). Derivates and the approximate differentiability (370).The class BMO (373). Weighted inequalities (374). Measures withthe doubling property (375). Sobolev derivatives (376). The area and coarea formulas and change of variables (379). Surface measures (383).The CalderSn-Zygmund decomposition (385).Exercises (386).Bibliographical and Historical CommentsReferencesAuthor IndexSubject Index

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