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Convergence Theorems for Lattice Group-valued Measures

  • ID: 3197929
  • Book
  • April 2015
  • Bentham Science Publishers Ltd
Convergence Theorems for Lattice Group-valued Measures explains limit and boundedness theorems for measures taking values in abstract structures. The book begins with a historical survey about these topics since the beginning of the last century, moving on to basic notions and preliminaries on filters/ideals, lattice groups, measures and tools which are featured in the rest of this text.

Readers will also find a survey on recent classical results about limit, boundedness and extension theorems for lattice group-valued measures followed by information about recent developments on these kinds of theorems and several results in the setting of filter/ideal convergence. In addition, each chapter has a general description of the topics and an appendix on random variables, concepts and lattices is also provided.

Thus readers will benefit from this book through an easy-to-read historical survey about all the problems on convergence and boundedness theorems, and the techniques and tools which are used to prove the main results. The book serves as a primer for undergraduate, postgraduate and Ph. D. students on mathematical lattice and topological groups and filters, and a treatise for expert researchers who aim to extend their knowledge base.
Note: Product cover images may vary from those shown
1. Historical Survey
1.1. Preliminaries
1.1.1. Topological Spaces and Groups
1.1.2. Boolean Algebras, Lattices and Related Structures
1.1.3. Set Functions
1.2. The Evolution of the Limit Theorems
1.2.1. The Sliding Hump
1.2.2. Vitali-Hahn-Saks-Nikodým, Schur and Dunford-Pettis Theorems
1.2.3. Finitely Additive, (s)-Bounded and (Uniformly) s-Additive Measures
1.2.4. Dieudonné, Grothendieck and Related Theorems
1.2.5. The Rosenthal Lemma
1.2.6. Limit Theorems for Finitely and s-Additive Measures and Matrix Theorems
1.2.7. The Drewnowski Theorem
1.2.8. (s)-Bounded Banach Space-Valued Measures
1.2.9. The Biting Lemma
1.2.10. Basic Matrix Theorems
1.2.11. Measures Defined on Algebras
1.2.12. Vector Lattice-Valued Measures
1.2.13. Measures Defined on Abstract Structures

2. Basic Concepts and Results
2.1. Filters and Ideals
2.1.1. Statistical Convergence and Matrix Methods
2.1.2. Basic Concepts and Properties of Ideals/Filters
2.1.3. Filter/Ideal Convergence
2.1.4. Almost Convergence
2.1.5. Filter Compactness
2.2. Filter Convergence in Lattice Groups
2.2.1. Basic Properties of Lattice Groups
2.2.2. Filter Convergence/Divergence
2.3. Lattice Group-Valued Measures
2.3.1. Main Properties of Measures
2.3.2. Countably Additive Restrictions
2.3.3. Carathéodory and Stone Extensions
2.3.4. Bounded Functions and Limits
2.3.5. M-Measures and their Extensions

3. Classical Limit Theorems in Lattice Groups
3.1. Convergence Theorems in the Global Sense
3.1.1. Uniform (s)-Boundedness and Related Topics
3.1.2. The Dieudonné Theorem
3.2. Construction of Integrals
3.2.1. Bochner-Type Integrals
3.2.2. Integrals with Respect to Optimal Measures
3.2.3. Ultrafilter Measures and Integrals
3.3. Further Limit Theorems
3.3.1. Brooks-Jewett Theorem
3.3.2. Dieudonné Theorem

3.4. Decomposition Theorems for (l)-Group-Valued Measures
3.4.1. Lebesgue-Type Decompositions
3.4.2. Sobczyk-Hammer-Type Decompositions
3.4.3. Yosida-Hewitt-Type Decompositions

4. Filter/Ideal Limit Theorems
4.1. Filter Limit Theorems in Lattice Groups
4.1.1. Schur-Type Theorems and Consequences
4.1.2. Other Nikodým and Brooks-Jewett-Type Theorems
4.1.3. Dieudonné-Type Theorems
4.1.4. The Uniform Boundedness Principle
4.1.5. The Basic Matrix Theorem
4.2. Filter Exhaustiveness and Convergence Theorems
4.2.1. Filter Exhaustiveness
4.2.2. Stone Extensions and Equivalence Results Between Limit Theorems
4.3. Modes of Continuity of Measures
4.3.1. Filter Continuity
4.3.2. Filter (a)-Convergence
4.3.3. Filter Weak Compactness and Weak Convergence of Measures
4.4. Topological Group-Valued Measures
4.4.1. Basic Properties
4.5. Filter Limit Theorems for Topological Group-Valued Measures
4.5.1. Schur-Type Theorems
4.5.2. Other Types of Limit Theorems
4.5.3. Limit Theorems for Positive Measures
4.5.4. Filter Exhaustiveness and Equivalence Results

General Discussion

Appendix

1.1. Random Variables
1.2. Concept, Lattice and Probability, by X. Dimitriou and C.P. Kitsos

References
Index
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