Advanced Integration Theory - Corneliu Constantinescu

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Résumé :
This book gives a general definition of the (abstract) integral, using the Daniell method. A most welcome consequence of this approach is the fact that integration theory on Hausdorff topological spaces appears simply to be a special case of abstract integration theory. The most important tool for the development of the abstract theory is the theory of vector lattices which is presented here in great detail. Its consequent application not only yields new insight into integration theory, but also simplifies many proofs. For example, the space of real-valued measures on a delta-ring turns out to be an order complete vector lattice, which permits a coherent development of the theory and the elegant derivation of many classical and new results. br/ The exercises occupy an important part of the volume. In addition to their usual role, some of them treat separate topics related to vector lattices and integration theory. br/ emAudience:/em This work will be of interest to graduate-level students and researchers with a background in real analysis, whose work involves (abstract) measure and integration, vector lattices, real functions of a real variable, probability theory and integral transforms.

Sommaire:
Suggestion to the Reader.- 0 Preliminaries.- Vector Lattices.- 1.1 Ordered Vector Spaces.- 1.2 Vector Lattices.- 1.3 Substructures, Quotients, Products.- 1.4 Bands and Orthogonality.- 1.5 Homomorphisms.- 1.6 The Order Dual of a Vector Lattice.- 1.7 Continuous Functionals.- 1.8 Order and Topology.- 1.9 Metric Spaces and Banach Spaces.- 1.10 Banach Lattices.- 1.11 Hilbert Lattices.- 1.12 Lattice Products.- Elementary Integration Theory.- 2.1 Riesz Lattices.- 2.2 Daniell Spaces.- 2.3 The Closure of a Daniell Space.- 2.4 The Integral for a Daniell Space.- 2.5 Systems of Sets, Step Functions, and Stone Lattices.- 2.6 Positive Measures.- 2.7 Closure, Completion, and Integrals for Positive Measure Spaces.- 2.8 Measurable Spaces and Measurability.- 2.9 Measurability versus Integrability.- 2.10 Stieltjes Functionals and Stieltjes Measures. Lebesgue Measure.- 3 Lp-Spaces.- 3.1 Classes modulo ?, and Convergence in Measure.- 3.2 The H?lder and Minkowski Inequalities and the Lp-Spaces.- 3.3 Lp-Spaces for 0< p< ?.- 3.4 Uniform integrability and the Generalized Lebesgue Convergence Theorem.- 3.5 Localization.- 3.6 Products and Lp?.- Real Measures.- 4.1 Nullcontinuous Functionals.- 4.2 Real Measures and Spaces of Real Measures.- 4.3 Integrals for Real Measures.- 4.4 Bounded Measures.- 4.5 Atomic and Atomless Measures.- The Radon-Nikodym Theorem. Duality.- 5.1 Absolute Continuity.- 5.2 The Theorem of Radon-Nikodym.- 5.3 Duality for Function Spaces.- 6 The Classical Theory of Real Functions.- 6.1 Functions of Locally Finite Variation.- 6.2 Real Stieltjes Measures.- 6.3 Absolutely Continuous Functions.- 6.4 Vitali?s Covering Theorem.- 6.5 Differentiable Functions.- 6.6 Spaces of Multiply Differentiable Functions.- 6.7 Riemann-Stieltjes Integrals.- Historical Remarks.- Name Index.-Symbol Index.