#### 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