幾何測度論

內(nèi)容概要

This book aims to fill the need for a comprehensive treatise on geo-metric measure theory. It contains a detailed exposition leading from the foundations of the theory to the most recent discoveries, including many results not previously published. It is intended both as a reference book for mature mathematicians and as a textbook for able students. The material of Chapter 2 can be covered in a first year graduate course on real analysis. Study of the later chapters is suitable preparation for re-search. Some knowledge of elementary set theory, topology, linear algebra and commutative ring theory is prerequisite for reading this book, but the treatment is selfcontained with regard to all those topics in multilinear algebra, analysis, differential geometry and algebraic topology which occur.

作者簡介

Herbert Federer was born on July 23, 1920, in Vienna. After emigrating to the US in 1938, he studied mathematics and physics at the University of California, Berkeley. Affiliated to Brown University, Providence since 1945, he is now Professor Emeritus there.
The major part of Professor Federer's scientific effort has been directed to the development of the subject of Geometric Measure Theory, with its roots and applications in classical geometry and analysis, yet in the functorial spirit of modern topology and algebra. His work includes more than thirty research papers published between 1943 and 1986, as well as this book.

書籍目錄

IntroductionCHAPTER ONE  Grassmann algebra 1.1. Tensor products 1.2. Graded algebras 1.3. The exterior algebra of a vectorspace 1.4. Alternating forms and duality 1.5. Interior multiplications 1.6. Simple m-vectors 1.7. Inner products 1.8. Mass and comass 1.9. The symmetric algebra of a vectorspace 1.10. Symmetric forms and polynomial functionsCHAPTER TWO   General measure theory 2.1. Measures and measurable sets  2.1.1. Numerical summation  2.1.2.-3. Measurable sets  2.1.4.-5. Measure hulls  2.1.6. Ulam numbers 2.2. Borel and Suslin sets  2.2.1. Borel families  2.2.2. -3. Approximation lay closed subsets  2.2.4. -5. Nonmeasurable sets  2.2.5. Radon measures  2.2.6. The space of sequences of positive integers  2.2.7. -9. Lipschitzian maps  2.2.10.-13. Suslin sets  2.2.14.-15. Borel and Baire functions  2.2.16. Separability of supports  2.2.17. Images of Radon measures 2.3 Measurable functions  2.3.1.-2. Basic properties  2.3.3.-7. Approximation theorems  2.3.8.-10. Spaces of measurable functions 2.4. Lebesgue integration  2.4.1.-5. Basic properties  2.4.6.-9. Limit theorems  2.4.10.-11. Integrals over subsets  2.4.12.-17. Lebesgue spaces  2.4.18. Compositions and image measures  2.4.19. Jensen''s inequality 2.5. Linear functionals  2.5.1. Lattices of functions  2.5.2.-6. Daniell integrals  2.5.7.-12. Linear functionals on Lebesgue spaces  2.5.13.-15. Riesz''s representation theorem  2.5.16. Curve length  ……CHAPTER THREE  RectifiabilityCHAPTER FOUR  Homological integration theoryCHAPTER FIVE  Applications to the calculus of variationsBibliographyGlossary of some standrd notationsList of basic notations defined in the textIndex

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