Integration And Modern Analysis

This textbook treats integration theory and its fascinating evolution throughout the past century, unifying the subject of real analysis using several central concepts, in particular, the notion of absoute continuity. Absolute continuity is essential in dealing with problems of switching limits and of understanding the powerful inverse relationship between integration and differentiation. It also leads to fundamental notions and results in terms of weak convergence of measures, a topic generally not treated in standard real variable texts.The work 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. The core chapters are followed by chapters of a topical nature, which illuminate the authors' intellectual vision of modern real analysis. These topics include weak convergence, the Riesz representation theorem, the Lebesgue differential theorem, and self-similar sets and fractals.Historical remarks, illuminating problems and examples, and appendices on functional analysis and Fourier analysis provide insight into the theory and its applications. The self-contained and fundamental coverage of the theories of integration, differentiation, and modern analysis make this text ideal for graduate students in the classroom setting.