Some New Types of Filter Limit Theorems for Topological Group-Valued Measures

• Published in: Real analysis exchange Vol. 39; no. 1; pp. 139 - 174
• Main Authors: Boccuto, Antonio, Dimitriou, Xenofon
• Format: Journal Article
• Language: English
• Published: East Lansing Michigan State University Press 01.01.2013
• Subjects: (free) filter; (uniform) \(\sigma\)-additivity; (uniform) continuity; (uniform) regularity; (uniform) S boundedness; 06E15; 22A05; 22A10; 28A12; 28A33; 28B10; 28B15; 28C15; 40A35; 40C05; 40G15; 41A35; 41A36; 54A20; 54E15; 54H11; block-respecting filter; Brooks-Jewett theorem; diagonal filter; Dieudonne theorem; equivalence between limit theorems; filter convergence; Frechet-Nikod\'ym topology; Nikodym convergence theorem; P-filter; sliding hump; Theorems; Theoretical mathematics; topological group; uniform filter exhaustiveness; Vitali-Hahn-Saks theorem
• ISSN: 0147-1937; 1930-1219

Some new types of limit theorems for topological group-valued measures are proved in the context of filter convergence for suitable classes of filters. We investigate (s) -boundedness, ...-additivity and regularity properties of topological group-valued measures. We consider also Schur-type theorems, using the sliding hump technique, and prove some convergence theorems in the particular case of positive measures. We deal with the notion of uniform filter exhaustiveness, by means of which we prove some theorems on existence of the limit measure, some other kinds of limit theorems and their equivalence, using known results on existence of countably additive restrictions of strongly bounded measures. Furthermore we pose some open problems. (ProQuest: ... denotes formulae/symbols omitted.)