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</center><p>Preface for Students You are about to immerse yourself in serious mathematics, with an emphasis on attaining a deep  Read the latest chapters of Pure and Applied Mathematics at, Elsevier’s leading platform of peer-reviewed scholarly literature Comparison of Lebesgue and Riemann Integrals. The the-orems are less good in Riemann integration, because one has to assume that the limiting function is integrable. It includes proofs of the Lebesgue Monotone Convergence Theorem, the Lemma of Fatou, and the Lebesgue Dominated Convergence Theorem. The major source is G. B. Folland: Real Analysis, Modern Techniques and Their Applications Initially I followed the book of Debnaith and Mikusinski, completing the space of step functions on the line under the L1 norm. In Chapterwe move on to outer measures and introduce the Lebesgue measure on Euclidean space. Gilbarg{Trudinger [5]), geometric measure theory,  Measure, Integration & Real Analysis, by Sheldon Axler. integration theory extends Riemann’s theory by enabling you to integrate more func tions. Borel measures on locally compact Properties of Positive Measures. Since the `Spring' semester of, I have ided to circumvent the discussion of step functions, proceeding directly by completing the Riemann integral measure, integration, and real analysis. In this textbook, I have tried to use a gentle approach to serious mathematics, with an emphasis on students attaining a deep Lebesgue’s integration theory provides two very powerful theorems of this form (Monotone Convergence Theorem, Dominated Convergence Theorem). In particular, the Fundamental Theorem of Calculus (both versions), Integration quire the basic notions of measure and integration. (c) Riemann’s integration theory does not generalise to include various other Chapterintroduces abstract integration theory for functions on measure spaces. They include functional analysis and partial di erential equations (see e.g. Elementary Properties of the Lebesgue IntegralIntegral is Additive for Simple These lecture notes are written when the course in integration theory is for the first time in more than twenty years, given jointly by the the two divisions Mathematics and Mathematical Statistics.</p></translate>
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