Integration And Modern Analysis

The subject of real analysis is unified in this modern text by several central concepts, in particular the notion of absolute continuity. Absolute continuity, a major underlying idea in formulating and proving the Fundamental Theorem of Calculus (FTC), the Lebesgue Dominated Convergence Theorem, and the Radon--Nikodym Theorem, is essential in dealing with problems of switching limits and of understanding the amazing inverse relationship between integration and differentiation (FTC). Also, it leads to fundamental notions and results in terms of weak convergence of measures, a topic generally not treated in standard real variable texts. The topics are examined in depth and in a way that is valuable not only from a theoretical point of view, but in a way that presents a sophisticated array of tools with which to deal with a host of difficult practical problems. Key features: Historical remarks and perspective on the development of real variables. Important role of Vitali, as responsible for many "firsts" in the subject as well as for "modern" proofs given almost 100 years ago * Covers all basic results in real variables, showing many possible generalizations and both standard and nonstandard examples * Comprehensive bibliography that contains both original works of the founding fathers of real analysis as well as relevant modern ones * Expanded material on distribution theory and the Riesz Representation Theorem to present a more unified theory of measure and integral * A careful treatment of symmetric perfect sets is used as a foundation for the introduction of fractals and analysis on them --- a subject that has emerged in recent years * An extensive array of exercises, ranging from the routine to more challenging problems * Comprehensive index. The book may be used not only for graduate courses in real analysis but also for advanced independent study, or as a source of problems for more advanced student projects.