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Since about 1915 integration theory has consisted of two separate branches: the abstract theory required by probabilists and the theory, preferred by analysts, that combines integration and topology. As long as the underlying topological space is reasonably nice (e.g., locally compact with countable basis) the abstract theory and the topological theory yield the same results, but for more compli cated spaces the topological theory gives stronger results than those provided by the abstract theory. The possibility of resolving this split fascinated us, and it was one of the reasons for writing this book. The unification of the abstract theory and the topological theory is achieved by using new definitions in the abstract theory. The integral in this book is de fined in such a way that it coincides in the case of Radon measures on Hausdorff spaces with the usual definition in the literature. As a consequence, our integral can differ in the classical case. Our integral, however, is more inclusive. It was defined in the book "C. Constantinescu and K. Weber (in collaboration with A.
List of contents
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 Hölder 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.
About the author
Karl Weber is an author specializing in business, social, and political topics.
