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.
目录
Introduction CHAPTER 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 functions CHAPTER 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 Rectifiability CHAPTER FOUR Homological integration theory CHAPTER FIVE Applications to the calculus of variations Bibliography Glossary of some standrd notations List of basic notations defined in the text Index
