Convergence Theorems For Lattice Group-Valued Measures


by

Antonio Boccuto, Xenofon Dimitriou

eISBN: 978-1-68108-009-3, 2015
ISBN: 978-1-68108-010-9



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Convergence Theorems for Lattice Group-valued Measures explains limit and boundedness theorems for measures taking va...[view complete introduction]

Table of Contents

Foreword

- Pp. i-ii (2)

Christos P. Kitsos

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Preface

- Pp. iii-iv (2)

Antonio Boccuto and Xenofon Dimitriou

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About the Authors

- Pp. v

Antonio Boccuto and Xenofon Dimitriou

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Historical Survey

- Pp. 3-139 (137)

Antonio Boccuto and Xenofon Dimitriou

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Basic Concepts and Results

- Pp. 140-262 (123)

Antonio Boccuto and Xenofon Dimitriou

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Classical Limit Theorems in Lattice Groups

- Pp. 263-358 (96)

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Filter/Ideal Limit Theorems

- Pp. 359-493 (135)

Antonio Boccuto and Xenofon Dimitriou

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General Discussion

- Pp. 494-498 (5)

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Appendix

- Pp. 499-508 (10)

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References

- Pp. 509-532 (24)

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Index

- Pp. 533-537 (5)

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Foreword

The eBook I am glad to read is a survey of the famous limit theorems for measures (Nikodým convergence theorem, Brooks-Jewett theorem, Vitali-Hahn-Saks theorem, Dieudonné convergence theorem, Schur convergence theorem). The first chapter seems to be the back bone of the eBook’s development. Not only it describes the development of the main theorems in the realm of convergence, but also provides a compact review of measures defined on algebras, vector latticevalued measures and measures defined on abstract structures. The use of these ideas is extensively described in Chapters 3 and 4. The historical development was enthusiastically approached since the second author was preparing his master’s thesis. Since then he worked consistently on the subject. Both authors are lovers of the historical development due to their Latin and Greek origin! Therefore the reader has the choice to appreciate an excellent piece of work on this area. The connection of Lattice Theory with Measure is explicitly described in this eBook and therefore the reader can also be addressed to Probability Theory. That is this eBook offers not only a strong background on limit theorems in Measure Theory, but also a solid theoretical insight into the Probability concepts. The norm of a measure, defined in Section 1.2, the definition of a measure on an algebra are essential tools to anybody working not only on Measure Theory but on Probability Theory as well. The next step, the definition of a measure defined on an abstract structure, needs more investigation in future work, while the authors cover completely the subject up to our days.

The definition of a Filter, defined firstly, and its dual notion of Ideal, defined later, are very nicely presented in Chapter 2. The relation between two Ideals is discussed in Section 2.1 as well as the Free Filters and P-Filters. These definitions and results are applied in Chapter 4. Being the authors consistent to their approach to limit theorems, they are extending Filters and Ideals with the corresponding limit theorems to Lattice Groups. Therefore a Lattice-Group-valued Measure is defined and the appropriate results are collected and presented. Nice examples on Filter Convergence in Lattice Groups help the reader to understand common ideas such as limsup or liminf through their development. The relation to Dedekind Complete Space is also discussed and related to Measure Theory. Therefore, I believe, the interested researcher has a compact, solid and rigorous presentation of Filters and Ideals.

The group with structure of lattice, known as (ℓ)-group, is what the authors investigate extensively in Chapter 3. The sense of Integration is very strictly presented under the light of Measure Theory. The convergence theorems for integrals are direct applications to Integration. The theoretical development of this Chapter is applied in Chapter 4, where a number of results is discussed under milder/weaker assumptions. Not only the limit theorems are presented but also interesting decomposition analogues for (ℓ)-group-valued measures are also discussed.

Chapter 4 is devoted to Filter (Ideal) Limit Theorems and their applications. Limit results and convergence theorems are presented in such a way the reader realizes that the authors are the grand masters of this subject. The Regularity of a Measure is discussed on any Dedekind Complete (ℓ)- Group. Topology is hidden everywhere and therefore also in Group-Valued Measures. This part is strongly related to the Preliminaries presented in Section 1.1, where the ideas of Topology, Measure and Banach Lattice are introduced.

The authors have collected more than 750 references on the subject. It is impressive not only for the extensively great number of references covering a wide variety of disciplines, but also for the fact that the authors refer to all of them inside the eBook.

I was glad when the authors asked me to write the preface. Then I realized that it was a hard work to go through this eBook. But I was eventually happy to realize that this excellent eBook covers the subject as well as possible. I did not have the chance to read such a compact review on the subject. I thank the authors for giving me the chance to read it.

Prof. Christos P. Kitsos
Department of Informatics
Technological Educational Institute of Athens
Chair of the ISI Committee on Risk Analysis
Greece


Preface

One of the topics of wide interest for several mathematicians, which has been successfully widely studied for more than a century, are the convergence and boundedness theorems for measures, in connection with properties of integrals, double sequences, matrix theorems and interchange of limits. Some related results in this area are the Banach-Steinhaus theorem in the operator setting and integration theory together with its fundamental properties. These topics have several applications in different branches of Mathematics, like for example topology, function spaces and approximation theory.

At the beginning, the case of σ-additive real-valued measures and integrals was treated, together with matrix theorems. These topics have been developed in the literature along several directions. Firstly, by considering not only countably additive, but also finitely additive measures and even set functions which are not necessarily finitely additive. Secondly, dealing with measures with values in abstract structures, like for instance Banach, uniform and locally convex spaces, topological and lattice groups, and so on. Thirdly, investigating measures defined on algebras satisfying suitable properties but which are not necessarily σ-algebras, or more abstract structures like for example MV-algebras, orthomodular posets, D-posets, minimal clans, which have several applications, for instance to quantum mechanics and multivalued logics.

To prove the main results about these topics, there are two types of techniques: the sliding hump or diagonal argument, which studies properties of the diagonal of an infinite matrix whose rows and columns are convergent, and the Baire category theorem. The sliding hump was known just at the beginning of the last century and was used for the proofs of the first fundamental results about limit theorems. The technique which uses the Baire category theorem is based on certain properties of Fréchet-Nikodým topologies. But this method, in general, is not adaptable in the finitely additive case. So, in most cases, it has been preferable to consider again the sliding hump method, which has been deeply studied in proving limit and boundedness theorems and also in matrix diagonal lemmas, which are very useful for these subjects. Furthermore, two procedures to relate the finitely additive case to the countably additive case have been investigated: the first deals with Stone-type σ-additive extensions of the original measures, and the second uses Drewnowski-type σ-additive restrictions of finitely additive measures on suitable σ-algebras.

The novelty of the research of the authors, which is exposed in Chapter 4, is to study limit theorems in the setting of filter convergence, which is an extension of convergence generated by matrix summability methods and includes as a particular case the statistical convergence, which is related with the filter of all subsets of the natural numbers having asymptotic density one. Note that, in general, it is impossible to expect analogous results corresponding to the classical case, even for σ-additive real-valued measures, because in general filter convergence is not inherited by subsequences. However it has been possible to prove several versions of limit, matrix and boundedness theorems as well as some results about different modes of continuity and convergence for measures, filter exhaustiveness (extending to the filter setting the concept of equicontinuity), continuity properties of the limit measure, weak filter/ideal compactness, and so on. The first chapter contains a historical survey of these topics since the beginning of last century. In Chapter 2 we deal with the basic concepts and tools used, like for instance filters/ideals, lattice group-valued measures, filter/ideal convergence in ( l )-groups, and present some fundamental tool, like for example the Maeda-Ogasawara-Vulikh representation theorem and the Stone Isomorphism technique. Chapter 3 contains several versions of limit and boundedness theorems for lattice group-valued measures and some applications to integrals. In the appendix we present an abstract approach on probability theory and random variables in connection with Boolean algebras, metric spaces, σ-additive extensions of finitely additive functions, various kinds of convergence in the lattice setting and tools which can have further developments, and we present some developments of the abstract notion of concept and some applications to Bioassays and related topics investigated by X. Dimitriou and C. P. Kitsos.

The eBook can be used both as a primer on limit theorems and filters/ideals and related topics, for postgraduate and Ph. D. students who want to explore these subjects and their beautifulness, and as a text for advanced researchers, since it exposes some new directions and results, shows some possibilities of further developments and ideas and includes also some open problems in the area.

This eBook includes several topics and developments of the research, started with the Ph. D. thesis of Dr. Xenofon Dimitriou, which was brilliantly discussed on 22th December 2011 under the supervision of Proffs. Nikolaos Papanastassiou and Antonio Boccuto at the National and Kapodistrian University of Athens.

The first author wants to dedicate the eBook to the loving memory of his parents. His father Giuliano died on 8th March 2011, while the authors were cooperating on the topics of the research exposed in this eBook. The second author wants to dedicate the eBook to his parents and to all who support him.

We want to thank Prof. Christos P. Kitsos for writing the foreword and Prof. Władysław Wilczyński for having translated from the Russian the papers by Doubrovsky, which the authors consulted for the preparation of Chapter 1. We thank also the Bentham Science Publishers, in particular Manager Hira Aftab and all her team, for their support and efforts. We also thank the referees for their remarks and suggestons, which improved the exposition of the eBook.

Acknowledgement

None declared.

Conflict Of Interest

The authors confirm that this eBook content has no conflict of interest.

Antonio Boccuto
Dipartimento di Matematica e Informatica
via Vanvitelli, 1
I-06123 Perugia
Italy
Email: antonio.boccuto@unipg.it

&

Xenofon Dimitriou
Department of Mathematics
University of Athens, Panepistimiopolis
Athens 15784
Greece
E-mails: xenofon11@gmail.com, dxenof@math.uoa.gr

List of Contributors

Author(s):
Antonio Boccuto
Dipartimento di Matematica e Informatica
via Vanvitelli, 1-06123 Perugia
Italy


Xenofon Dimitriou
Department of Mathematics
University of Athens
Panepistimiopolis, Athens 15784
Greece




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