Convergence Theorems for Lattice Group-Valued Measures

This chapter contains a historical survey about limit and boundedness theorems for measures since the beginning of the last century. In these kinds of theorems, there are two substantially different methods of proofs: the sliding hump technique and the use of the Baire category theorem. We deal with Vitali-Hahn-Saks, Brooks-Jewett, Nikodým convergence and boundedness theorems, and we consider also some related topics, among which Hahn-Schur-type theorems and some other kind of matrix theorems, the uniform boundedness principle and some (weak) compactness properties of spaces of measures. In this context, the Rosenthal lemma, the biting lemma and the Antosik-Mikusiński-type diagonal lemmas play an important role. We consider the historical evolution of convergence and boundedness theorems for σ- additive, finitely additive and non-additive measures, not only real-valued and defined on σ-algebras, but also defined and/or with values in abstract structures.

In this chapter we recall the fundamental concepts, tools and results which will be used throughout the book, that is filters/ideals, filter/ideal convergence, lattice groups, Riesz spaces and properties of (l)-group-valued measures, and some related fundamental techniques in this setting, like for instance different kinds of convergence, the Fremlin lemma, the Maeda-Ogasawara-Vulikh representation theorem, the Stone Isomorphism technique and the existence of suitable countably additive restrictions of finitely additive strongly bounded measures. We will prove some main properties of filter/ideal convergence and of lattice group-valued measures.

We consider several versions of limit theorems for lattice group-valued measures, in which both pointwise convergence of the involved measures and the notions of σ-additivity, (s)-boundedness, regularity, are given in the global sense, that is with respect to a common regulator. We present the construction of some kinds of integrals in the vector lattice context and some Vitali and Lebesgue theorems. Successively we prove some other kinds of limit theorems, in which the main properties of the measures are considered in the classical like sense. Finally, we give different types of decomposition theorems for lattice group-valued measures.

We present recent versions of limit and boundedness theorems in the setting of filter convergence, for measures taking values in lattice or topological groups, in connection with suitable properties of filters. Some results are obtained by applying classical versions to a subsequence, indexed by a family of the involved filter: in this context, an essential role is played by filter exhaustiveness. We give also some basic matrix theorems for lattice group-valued double sequences, in the setting of filter convergence. We give some modes of continuity for measures with respect to filter convergence, some comparisons between filter exhaustiveness and filter (α)- convergence of measure sequences and some weak filter Cauchy-type conditions, in connection with integral operators.

We give a summary of the main concepts, ideas, tools and results of Chapters 2,3,4. In Chapter 2 we have presented the basic notions and results about filters/ideals, statistical and filter/ideal convergence, both in the real case and in abstract structures. In Chapter 3 we have given the classical limit theorems and the Nikodým boundedness theorem for lattice group-valued measures, different types of decompositions and the construction of optimal and Bochner-type integrals in the lattice group setting. In Chapter 4 we have proved different versions of Schur, Brooks-Jewett, Vitali-Hahn- Saks, Dieudonné, Nikodým convergence and boundedness theorems in the setting of filter convergence for lattice or topological group-valued measures, and also some different results on modes of continuity, filter continuous convergence, filter weak compactness and filter weak convergence of measures.

We present an abstract approach on probability measures, events and random variables, involving in particular lattice theory, distance functions, σ-additive extensions of finitely additive functions, some kinds of convergences in the lattice setting, which can be considered even in more abstract contexts. Furthermore we pose some open problems.