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.