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基本説明
This textbook begins with the fundamentals of classical real variables and leads to Lebesgue's definition of the integral, the theory of integration and the structure of measures in a measure theoretical format. Features: Fascinating historical commentary interwoven into the exposition. Hundreds of problems from routine to challenging.
Full Description
This book is both a text and a paean to twentieth-century real variables, measure theory, and integration theory. As a text, the book is aimed at graduate students. As an exposition, extolling this area of analysis, the book is necessarily limited in scope and perhaps unnecessarily unlimited in id- syncrasy. More than half of this book is a fundamental graduate real variables course as we now teach it. Since there are excellent textbooks that generally cover the course material herein, part of this Preface renders an apologia for our content, presentation, and existence. The following section presents our syllabus properly liberated from too many demands. Subsequent sections dealwithoutline,theme,features,andtherolesofFourieranalysisandVitali, respectively. Mathematics is a creativeadventure drivenby beauty, structure, intrinsic mathematicalproblems,extrinsicproblemsfromengineeringandthesciences, and serendipity. This book treats integration theory and its fascinating creation through the past century. What about the rest of our title? "Analysis"is many subjects to many mathematicians."Modern analysis" is hardly a constraint for a single volume such as ours; one can argue the opposite. Di?erentiation and integrationare still the essence of analysis,and, along with "integration", the title could very well have included the word "di?erentiation" because of our emphasis on it. Guided by the creativity of mathematics,ourtitleismeanttoassertthatthetechnologywehaverecorded is a basis for many of the analytic adventures of our time. Syllabus We shall outline the material we have used in teaching a ?rst-year graduate course in real analysis. Sometimes a student will take only the ?rst semester of this two-semester sequence.
Contents
Classical Real Variables.- Lebesgue Measure and General Measure Theory.- The Lebesgue Integral.- The Relationship between Differentiation and Integration on.- Spaces of Measures and the Radon#x2013;Nikodym Theorem.- Weak Convergence of Measures.- Riesz Representation Theorem.- Lebesgue Differentiation Theorem on.- Self-Similar Sets and Fractals.- Functional Analysis.- Fourier Analysis.
