Convergence Theorems for Lattice Group-Valued Measures

Convergence Theorems for Lattice Group-Valued Measures

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 ...
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Basic Concepts and Results

Pp. 140-262 (123)

Antonio Boccuto and Xenofon Dimitriou

Abstract

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.

Keywords:

(s)-bounded measure, (Uniform) asymptotic density, absolutely continuous measure, additive measure, almost convergence, block-respecting filter, Carathéodory extension, diagonal filter, filter compactness, filter divergence, filter, filter/ideal convergence, Fremlin Lemma, ideal, lattice group, Maeda-Ogasawara-Vulikh theorem, matrix method, P-filter, regular measure, Stone extension.

Affiliation:

Dipartimento di Matematica e Informatica via Vanvitelli, 1 I-06123 Perugia Italy.