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Advanced Integration Theory

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...

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Bibliographic Details
Main Authors: Constantinescu, Corneliu, Filter, Wolfgang (Author), Weber, Karl (Author)
Format: eBook
Language:English
Published: Dordrecht Springer Netherlands 1998, 1998
Edition:1st ed. 1998
Series:Mathematics and Its Applications
Subjects:
Measure Theory
Mathematical Analysis
Functions Of Real Variables
Probability Theory
Integral Transforms And Operational Calculus
Algebra
Real Functions
Order, Lattices, Ordered Algebraic Structures
Measure And Integration
Probabilities
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table 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

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