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基本説明
In the setting of a general measure space, every concepts is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details.
Full Description
This book presents a unified treatise of the theory of measure and integration. In the setting of a general measure space, every concept is defined precisely and every theorem is presented with a clear and complete proof with all the relevant details. Counter-examples are provided to show that certain conditions in the hypothesis of a theorem cannot be simply dropped.The dependence of a theorem on earlier theorems is explicitly indicated in the proof, not only to facilitate reading but also to delineate the structure of the theory. The precision and clarity of presentation make the book an ideal textbook for a graduate course in real analysis while the wealth of topics treated also make the book a valuable reference work for mathematicians.
Contents
Measure and Outer Measure; Regularity of Measures; Measurable Mappings; Completion of a Measure Space; Convergence Almost Everywhere; Almost Uniform Convergence; Convergence in Measure; Integration with Respect to a Measure; Generalized Convergence Theorems for Integrals; Signed Measures; Absolute Continuity of a Measure with Respect to Another; Monotone Functions and Functions of Bounded Variation on R; Absolutely Continuous Functions; Convex Functions, Differentiation of an Indefinite Integral; Banach Spaces; Lp Spaces for p in (0, ); Bounded Linear Functionals; Integration on a Locally Compact Hausdorff Space; Extension of Additive Set Functions to Measures; Lebesgue-Stieltjes Measure Space; Product Measure Spaces; Convolution of Functions; Integration with Respect to Lebesgue Measure on Euclidean Spaces; Integral and Linear Transformations of the Integral; Hardy-Littlewood Maximal Theorem; Lebesgue Differentiation Theorem; Change of Variable of Integration by Differentiable Transformations; Hausdorff Measures on Euclidean Spaces; Hausdorff Dimensions; Transformation of Hausdorff Measures.
