Filter/Ideal Limit Theorems
Pp. 359-493 (135)
Antonio Boccuto and Xenofon Dimitriou
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.
(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.
Dipartimento di Matematica e Informatica via Vanvitelli, 1 I-06123 Perugia Italy.