Research Article
Invariant Statistical Convergence of Sequences of Sets with
respect to a Modulus Function
Nimet Pancaroglu and Fatih Nuray
Department of Mathematics, Afyon Kocatepe University, Afyonkarahisar, Turkey
Correspondence should be addressed to Fatih Nuray; fnuray@aku.edu.tr
Received 15 December 2013; Accepted 21 February 2014; Published 23 March 2014
Academic Editor: Irena Rach˚ unkov´ a
Copyright © 2014 N. Pancaroglu and F. Nuray. is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
We introduce and study the concept of invariant convergence for sequences of sets with respect to modulus function and give
some inclusion relations.
1. Introduction
e concept of statistical convergence for sequences of real
numbers was introduced by Fast [1] and studied by
ˇ
Sal´ at [2]
and others. Let ⊆ N and
= { ≤ : ∈ }. en the
natural density of is defined by () = lim
−1
|
|, if the
limit exists, where |
| denotes the cardinality of
.
A sequence = (
) complex numbers is said to be
statistically convergent to if, for each >0,
lim
1
{ ≤ :
−
≥ }
= 0. (1)
Convergence concept for sequences of set had been
studied by Beer [3], Aubin and Frankowska [4], and Baronti
and Papini [5]. e concept of statistical convergence of
sequences of set was introduced by Nuray and Rhoades [6] in
2012. Ulusu and Nuray [7] introduced the concept of Wijsman
lacunary statistical convergence of sequences of set. Similarly,
the concepts of Wijsman invariant statistical and Wijsman
lacunary invariant statistical convergence were introduced by
Pancaroglu and Nuray [8] in 2013.
A function : [0, ∞) → [0, ∞) is called a modulus, if
(1) () = 0 if and if only if =0;
(2) ( + ) ≤ () + ();
(3) is increasing;
(4) is continuous from the right at 0.
A modulus may be unbounded (e.g., () =
, 0<<
1) or bounded (e.g., () = /( + 1)).
Modulus function was introduced by Nakano [9] in 1953.
Ruckle [10] used in idea of modulus function to construct
a class of FK spaces. Consider
() = { = (
):
∞
∑
=1
(
) < ∞} . (2)
e space () is closely related to the space ℓ
1
which is a
() space with () = , for all real ≥0.
Maddox [11] defined the following spaces by using a
modulus function :
0
() = { ∈ : lim
→∞
1
∑
=1
(
) = 0} ,
() = { ∈ : lim
→∞
1
∑
=1
(
−
)=0 for some } ,
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 818020, 5 pages
http://dx.doi.org/10.1155/2014/818020