Advanced Integration Theory

By: Corneliu Constantinescu, Alexia Sontag, Wolfgang Filter

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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. 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. Audience: 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.

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Preface	p. vii
Acknowledgements	p. x
Introduction	p. 1
Suggestions to the Reader	p. 5
Preliminaries	p. 7
Vector Lattices	p. 21
Ordered Vector Spaces	p. 21
Vector Lattices	p. 38
Substructures, Quotients, Products	p. 58
Bands and Orthogonality	p. 79
Homomorphisms	p. 95
The Order Dual of a Vector Lattice	p. 126
Continuous Functionals	p. 142
Order and Topology	p. 167
Metric Spaces and Banach Spaces	p. 197
Banach Lattices	p. 236
Hilbert Lattices	p. 260
Lattice Products	p. 272
Elementary Integration Theory	p. 279
Riesz Lattices	p. 279
Daniell Spaces	p. 293
The Closure of a Daniell Space	p. 308
The Integral for a Daniell Space	p. 330
Systems of Sets, Step Functions, and Stone Lattices	p. 340
Positive Measures	p. 352
Closure, Completion, and Integrals for Positive Measure Spaces	p. 371
Measurable Spaces and Measurability	p. 386
Measurability versus Integrability	p. 393
Stieltjes Functionals and Stieltjes Measures. Lebesgue Measure	p. 416
L[superscript p]-Spaces	p. 447
Classes modulo [mu] and Convergence in Measure	p. 447
The Holder and Minkowski Inequalities and the L[superscript p]-Spaces	p. 466
L[superscript p]-Spaces for 0 [ p [ [infinity]	p. 490
Uniform integrability and the Generalized Lebesgue Convergence Theorem	p. 506
Localization	p. 524
Products and L[superscript [infinity]]	p. 542
Real Measures	p. 549
Nullcontinuous Functionals	p. 549
Real Measures and Spaces of Real Measures	p. 559
Integrals for Real Measures	p. 578
Bounded Measures	p. 608
Atomic and Atomless Measures	p. 619
The Radon-Nikodym Theorem. Duality	p. 637
Absolute Continuity	p. 637
The Theorem of Radon-Nikodym	p. 656
Duality for Function Spaces	p. 663
The Classical Theory of Real Functions	p. 681
Functions of Locally Finite Variation	p. 681
Real Stieltjes Measures	p. 711
Absolutely Continuous Functions	p. 724
Vitali's Covering Theorem	p. 736
Differentiable Functions	p. 760
Spaces of Multiply Differentiable Functions	p. 781
Riemann-Stieltjes Integrals	p. 792
Historical Remarks	p. 805
Name Index	p. 808
Subject Index	p. 810
Symbol Index	p. 849
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Advanced Integration Theory

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