Filter/Ideal Limit Theorems

Abstract

We present recent versions of limit and boundedness theorems in the setting of filter convergence, for measures taking values in lattice or topological groups, in connection with suitable properties of filters. Some results are obtained by applying classical versions to a subsequence, indexed by a family of the involved filter: in this context, an essential role is played by filter exhaustiveness. We give also some basic matrix theorems for lattice group-valued double sequences, in the setting of filter convergence. We give some modes of continuity for measures with respect to filter convergence, some comparisons between filter exhaustiveness and filter (α)- convergence of measure sequences and some weak filter Cauchy-type conditions, in connection with integral operators.

Keywords: (Filter) continuous measure, Banach-Steinhaus theorem, basic matrix theorem, block-respecting filter, Brooks-Jewett theorem, Diagonal filter, Dieudonné theorem, Drewnowski theorem, equivalence, filter (α)-convergence, filter exhaustiveness, filter limit theorem, filter weak compactness, filter weak convergence, Nikodým boundedness theorem, Nikodým convergence theorem, Pfilter, Schur theorem, topological group, Vitali-Hahn-Saks theorem.