TY - BOOK ID - 4799067 TI - Integration and Modern Analysis AU - Benedetto, John J. AU - Czaja, Wojciech. PY - 2009 SN - 9780817646561 9780817643065 0817643060 9786613562098 0817646566 1280384174 PB - Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, DB - UniCat KW - Mathematics. KW - Analysis. KW - Functions of a Complex Variable. KW - Measure and Integration. KW - Global analysis (Mathematics). KW - Functions of complex variables. KW - Mathématiques KW - Analyse globale (Mathématiques) KW - Fonctions d'une variable complexe KW - Mathematical analysis KW - Functions of real variables KW - Integration, Functional KW - Integrals, Generalized KW - Measure theory KW - Electronic books. -- local. KW - Functions of real variables. KW - Integrals, Generalized. KW - Integration, Functional. KW - Mathematical analysis. KW - Measure theory. KW - Engineering & Applied Sciences KW - Applied Mathematics KW - Lebesgue measure KW - Measurable sets KW - Measure of a set KW - 517.1 Mathematical analysis KW - Functional integration KW - Real variables KW - Analysis (Mathematics). KW - Algebraic topology KW - Measure algebras KW - Rings (Algebra) KW - Functional analysis KW - Calculus, Integral KW - Functions of complex variables KW - Analysis, Global (Mathematics) KW - Differential topology KW - Geometry, Algebraic UR - https://www.unicat.be/uniCat?func=search&query=sysid:4799067 AB - A paean to twentieth century analysis, this modern text has several important themes and key features which set it apart from others on the subject. A major thread throughout is the unifying influence of the concept of absolute continuity on differentiation and integration. This leads to fundamental results such as the Dieudonné–Grothendieck theorem and other intricate developments dealing with weak convergence of measures. Key Features: * Fascinating historical commentary interwoven into the exposition; * Hundreds of problems from routine to challenging; * Broad mathematical perspectives and material, e.g., in harmonic analysis and probability theory, for independent study projects; * Two significant appendices on functional analysis and Fourier analysis. Key Topics: * In-depth development of measure theory and Lebesgue integration; * Comprehensive treatment of connection between differentiation and integration, as well as complete proofs of state-of-the-art results; * Classical real variables and introduction to the role of Cantor sets, later placed in the modern setting of self-similarity and fractals; * Evolution of the Riesz representation theorem to Radon measures and distribution theory; * Deep results in modern differentiation theory; * Systematic development of weak sequential convergence inspired by theorems of Vitali, Nikodym, and Hahn–Saks; * Thorough treatment of rearrangements and maximal functions; * The relation between surface measure and Hausforff measure; * Complete presentation of Besicovich coverings and differentiation of measures. Integration and Modern Analysis will serve advanced undergraduates and graduate students, as well as professional mathematicians. It may be used in the classroom or self-study. ER -