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.
Related Books
Boundary Element Methods for Heat Transfer with Phase Change Problems: Theory and Application
Advanced Calculus - Fundamentals of Mathematics
Advanced Numerical Methods for Complex Environmental Models: Needs and Availability
An Introduction to Sobolev Spaces
An Introduction to Trigonometry and its Applications
Analysis and Optimal Control of Phase-Field Transition System: Fractional Steps Methods
Applied Computational Mathematics in Social Sciences
Calculus: The Logical Extension of Arithmetic
Complex Analyses in Engineering, Science and Technology
Differential and Integral Calculus Theory and Cases