商品描述
The final section of each chapter in Part I presents problems that are integral to each chapter, the majority of which consist of auxiliary results, extensions of the theory, examples, and counterexamples. Problems which are highly theoretical have accompanying hints. The last section of each chapter of Part II consists of Additional Propositions containing auxiliary and complementary results. The entire book contains collections of suggested readings at the end of each chapter in order to highlight alternate approaches, proofs, and routes toward additional results. This second edition adds a new discussion on probability measures, some of which are scattered among proposed problems in Part I and all of them summarized in the Appendix to Part I. Chapters on decomposition of measures and representation theorems include substantially more material. A comprehensive discussion on the Cantor-Lebesque measure can be found in problems 7.15 and 7.16. Rajchman measures have been considered in Problems 7.17 and 7.18. There is a new subsection on Borel regular measures on topological spaces in Section 12.4.
商品描述(中文翻譯)
這本教科書在其方法上是經典的，經過深思熟慮的設計，分為兩個部分。第一部分旨在為一學期的研究生入門課程提供測度理論，提出一種「抽象」的方法來探討測度和積分，其中經典的具體案例——勒貝格測度和勒貝格積分被呈現為一般理論的一個重要特例。第一部分也可能對滿足先決條件的高年級本科生可及，這些先決條件包括分析學的入門課程、線性代數（僅限第5章）和初等集合論。文本的第二部分則更為進階，針對更有經驗的讀者。這部分的內容旨在涵蓋另一個學期的研究生課程，接續第一門課程，處理拓撲空間中的測度和積分。這本書的先決條件相對簡單，旨在滿足當代數學學生在測度理論課程中的需求，並且也對更廣泛的學生群體（如統計學、經濟學、工程學和物理學的學生）可及。
第一部分每章的最後一節呈現與每章密切相關的問題，其中大多數由輔助結果、理論的擴展、例子和反例組成。高度理論性的問題附有提示。第二部分每章的最後一節包含附加命題，提供輔助和補充結果。整本書在每章結尾都有建議閱讀的資料，以突顯替代的方法、證明和通往額外結果的路徑。這第二版新增了對概率測度的討論，其中一些散佈在第一部分的提議問題中，所有內容在第一部分的附錄中進行總結。關於測度的分解和表示定理的章節包含了更多的材料。關於Cantor-勒貝格測度的全面討論可以在問題7.15和7.16中找到。Rajchman測度在問題7.17和7.18中被考慮。第12.4節中有關拓撲空間的Borel正則測度的新子部分。
作者簡介
Carlos S. Kubrusly is a professor in the electrical engineering department at the Catholic University of Rio de Janeiro. His current area of research is in operator theory and functional analysis. Over the years, the results of his Work have been published in over 40 journals, 6 monographs/textbooks, and 2 contributed volumes. From 1992-1998 Carlos Kubrusly was the editor-in-chief of the Computational and Applied Mathematics journal which was then co-published with Birkhäuser Boston.
作者簡介(中文翻譯)
Carlos S. Kubrusly 是里約熱內盧天主教大學電機工程系的教授。他目前的研究領域為算子理論和泛函分析。多年來，他的研究成果已發表在超過40本期刊、6本專著/教科書以及2本合編書籍中。從1992年到1998年，Carlos Kubrusly 擔任《計算與應用數學》期刊的主編，該期刊當時與Birkhäuser Boston共同出版。
