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.
Keywords: (D)-convergence, Baire category theorem, Bochner integral,
decomposition, Drewnowski technique, filter exhaustiveness, filter, filter/ideal
convergence, Fremlin lemma, Ideal, lattice group, limit theorem, Maeda-Ogasawara-
Vulikh theorem, optimal integral, order convergence, positive regular property, Stone
Isomorphism technique, topological group, ultrafilter measures, uniform boundedness
theorem.
