On real-valued measures of statistical type and their applications to statistical convergence

“…Statistical convergence was introduced by H. Fast [9] and H. Steinhaus [16], which is a generalization of the usual notion of convergence. It is doubtless that the study of statistical convergence and its various generalizations has become an active research area [2,3,7,17,18]. In particular, P. Kostyrko, T.Šalát and W. Wilczynski [11] introduced two interesting generalizations of statistical convergence by using the notion of ideals of subsets of positive integers, which were named as I and I * -convergence, and studied some properties of I and I * -convergence in metric spaces.…”

Section: Introductionmentioning

confidence: 99%

On I-quotient mappings and I-cs'-networks under a maximal ideal

Zhou

2020

Appl. Gen. Topol.

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<p>Let I be an ideal on N and f : X → Y be a mapping. f is said to be an I-quotient mapping provided f−1(U) is I-open in X, then U is I-open in Y . P is called an I-cs′-network of X if whenever {xn}n∈N is a sequence I-converging to a point x ∈ U with U open in X, then there is P ∈ P and some n0 ∈ N such that {x, xn0} ⊆ P ⊆ U. In this paper, we introduce the concepts of I-quotient mappings and I-cs′-networks, and study some characterizations of I-quotient mappings and I-cs′- networks, especially J -quotient mappings and J -cs′-networks under a maximal ideal J of N. With those concepts, we obtain that if X is an J -FU space with a point-countable J -cs′-network, then X is a meta-Lindelöf space.</p>

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“…A non-negative finitely additive measure µ defined on the collection 2 N of all subsets of N is said to be a statistical measure if µ(N) = 1 and µ({k}) = 0 for all k ∈ N. Evidently, a statistical measure cannot be countably additive. Statistical measures were introduced in [5,6,1] and extensively studied in [7]. The filter generated by a statistical measure µ is the collection F µ of those subsets A ⊂ N for which µ(A) = 1.…”

Section: Introductionmentioning

confidence: 99%

Conglomerated filters, statistical measures, and representations by ultrafilters

Kadets¹,

Seliutin²

2020

Preprint

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Using a new concept of conglomerated filter we demonstrate in a purely combinatorial way that none of Erdös-Ulam filters or summable filters can be generated by a single statistical measure and consequently they cannot be represented as intersections of countable families of ulrafilters. Minimal families of ultrafilters and their intersections are studied and several open questions are discussed.

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On real-valued measures of statistical type and their applications to statistical convergence

Cited by 9 publications

References 13 publications

A functional characterization of measures and the Banach–Ulam problem

A functional characterization of measures and the Banach–Ulam problem

On I-quotient mappings and I-cs'-networks under a maximal ideal

Conglomerated filters, statistical measures, and representations by ultrafilters

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