This book covers Lebesgue integration and its generalizations from Daniell's point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main purpose of measure theory. From this point of view Lebesgue's integral can be had as a rather straightforward, even simplistic, extension of Riemann's integral; and its aims, definitions, and procedures can be motivated at an elementary level. The notion of measurability, for example, is suggested by Littlewood's observations rather than being conveyed authoritatively through definitions of Sigma-algebras, and good-cut-conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner. The approach taken provides the additional benefit of cutting the labor in half. The use of seminorms, ubiquitous in modern analysis, speeds things up even further. The book is intended for the reader who has some experience with proofs, a beginning graduate student for example. It will as well be useful to the advanced mathematician who is confronted with situations -- such as stochastic integration -- where the set-measuring approach to integration does not work.

Preface Chapter I: Review I.1 Introduction I.2 Notation I.3 The Theorem of Stone-Weierstra? I.4 The Riemann Integral I.5 An Integrability Criterion I.6 The Permanence Properties I.7 Seminorms Chapter II: Extension of the Integral II.1 Sigma-additivity II.2 Elementary Integrals II.3 The Daniell Mean II.4 Negligible Functions and Sets II.5 Integrable Functions II.6 Extending the Integral II.7 Integrable Sets II.8 Example of a Non-Integrable Function Chapter III: Measurability III.1 Littlewood's Principles III.2 The Permanence Properties III.3 The Integrability Criterion III.4 Measurable Sets III.5 Baire and Borel Functions III.6 Further Properties of Daniell's Mean III.7 The Procedures of Lebesgue and Carath?ory Chapter IV: The Classical Banach Spaces IV.1 The p-norms IV.2 The Lp-spaces IV.3 The Lp-spaces IV.4 Linear Functionals IV.5 The Dual of Lp IV.6 The Hilbert space L2 Chapter V: Operations on Measures V.1 Products of Elementary Integrals V.2 The Theorems of Fubini and Tonelli V.3 An Application: Convolution V.4 An Application: Marcinkiewicz Interpolation V.5 Signed Measures V.6 The Space of Measures V.7 Measures with Densities V.8 The Radon-Nikodym Theorem V.9 An Application: Conditional Expectation V.10 Differentiation Appendix A: Answers to Selected problems References Index of Notations Index

등록된 사용후기

사용후기 쓰기 새 창 더보기

사용후기가 없습니다.

등록된 상품문의

상품문의 쓰기새 창 더보기

상품문의가 없습니다.

배송정보

교환/반품 정보