General Discussion

Abstract

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.