Real analysis: modern techniques and their applications
This book covers the subject matter that is central to mathematical analysis: measure & integration theory, some point set topology, & rudiments of functional analysis. Also, a number of other topics are developed to illustrate the uses of this core material in important areas of mathematics & to introduce readers to more advanced techniques. Some of the material presented has never appeared outside of advanced monographs & research papers, or been readily available in comparative texts. About 460 exercises, at varying levels of difficulty, give readers practice in working with the ideas presented here.
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Decomposition and Differentiation of Measures 1 Signed Measures
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a-algebra a-finite absolutely continuous algebra Banach space Borel measure Borel set bounded called Cauchy Cc(X closed sets compact Hausdorff space compact sets compactification complete complex measure contains continuous functions Corollary define denote dense differential disjoint union distribution dominated convergence theorem equivalent example exists finite measure follows Fourier transform Haar measure Hausdorff space hence implies inequality intersection isomorphic LCH space Lebesgue measure Lemma linear functional linear map locally compact measurable function measure space metric space monotone convergence theorem Moreover neighborhood nonempty nonnegative normed vector space notation open sets operator outer measure pointwise polynomial Proo proof of Theorem Proposition prove Radon measure random variables regular result satisfies Section semifinite seminorms sequence signed measure simple functions Suppose surjective theory topological space topology uniformly unique weak x e SC