Lacunary Invariant Statistical Convergence of Sequences of Sets with Respect to a Modulus Function

In this paper, we introduce and study the concept of lacunary invariant convergence for sequences of sets with respect to modulus function f and give some inclusion relations.


Introduction
The concept of statistical convergence for sequences of real numbers was introduced by Fast [1] and studied by Šalát [2] and others.Let  ⊆ N and   = { ≤  :  ∈ }.Then 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, Convergence concept for sequences of set had been studied by Beer [3], Aubin and Frankowska [4], and Baronti and Papini [5].The 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.
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 The 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 : where  is space of all complex sequences.Later, Connor [12] extended his definition by replacing the Cesaro matrix with an arbitrary nonnegative matrix summability method  = (   ) as follow:
The mappings  are assumed to be one-to-one such that   () ̸ =  for all positive integers  and , where   () denotes the th iterate of the mapping  at .Thus,  extends the limit functional on , the space of convergent sequences, in the sense that () = lim , for all  ∈ .In case  is translation mapping () =  + 1, the  mean is often called a Banach limit and   , the set of bounded sequences all of whose invariant means are equal, is the set of almost convergent sequences.
It can be shown that where, Nuray and Savas ¸ [13] defined the following sequence spaces by using a modulus function  and a nonnegative regular matrix  = (  ): Definition 1 (see [14]).A set  of positive integers is said to have uniform invariant density of zero if uniformly in .
By using uniform invariant density, the following definition was given.
Let (, ) be a metric space.For any point  ∈  and nonempty subset  of , we define the distance from  to  by The concept of Wijsman convergence was introduced by Wijsman [7] as follows.
Let (, ) be a metric space.For any nonempty closed subsets ,   ⊆ , we say that the sequence for each  ∈ .This will be denoted by  − lim   = .
Convergence concept for sequences of set had been studied by Beer [3], Aubin and Frankowska [4], and Baronti and Papini [5].The concepts of Wijsman statistical convergence and Wijsman strong Cesaro summability were introduced by Nuray and Rhoades [6] as follows.
Let (, ) be a metric space.For any nonempty closed subsets ,   ⊆ , the sequence {  } is said to be Wijsman strongly Cesaro summable to  if, for each  ∈ , Let (, ) be a metric space.For any nonempty closed subsets ,   ⊆ , the sequence {  } is said to be Wijsman statistically convergent to  if, for  > 0 and each  ∈ , In this case we write  − lim    =  or   → ().

Main Result
The purpose of this paper is, by using a modulus function, to introduce and study new sequence spaces of sequences of sets.The following three definitions were given in [8].
Definition 3. Let (, ) be a metric space.For any nonempty closed subsets ,   ⊆ , we say that the sequence uniformly in .
In this case, we write   → (  ) and the set of all Wijsman invariant convergent sequences of sets will be denoted   .Definition 4. Let (, ) be a metric space.For any nonempty closed subsets ,   ⊆ , we say that the sequence {  } is Wijsman strongly invariant convergent to , if for each  ∈ , uniformly in .
In this case, we write   → ([  ]) and the set of all Wijsman strongly invariant convergent sequences of sets will be denoted [  ].In this case, we write   → (  ) and the set of all Wijsman invariant statistically convergent sequences of sets will be denoted   .
Let (, ) be metric space.For any nonempty closed subsets ,   ⊆  and  ∈ , we define the sequences of sets space [  ] ∞ as follows: Now, by using a modulus function, we introduce the following new sequence spaces of sequences of sets.Definition 6.Let (, ) be a metric space and  be a modulus function.For any nonempty closed subsets ,   ⊆ , we say that the sequence {  } is Wijsman strongly invariant convergent to  with respect to the modulus , if, for each  ∈ , uniformly in .
In this case we write   → ([  ()]) and the set of all Wijsman strongly invariant convergent sequences of sets with respect to the modulus  will be denoted [  ()].
Theorem 7. Let (, ) be a metric space.For any nonempty closed subsets ,   ⊆ .Then Hence, {  } is strongly invariant convergent to  with respect to the modulus function .
(iii) This is an immediate consequence of (i) and (ii).This completes the proof of the theorem.Lemma 10 (see [15]).Let  be a modulus function.Let  > 0 be given constant.Then, there is a constant  > 0 such that () >  (0 <  < ).
Theorem 11.Let {  } be a bounded sequence and  be a modulus function.Then {  } is Wijsman strongly invariant convergent to  with respect to modulus  if and only if {  } is Wijsman strongly invariant convergent to ; that is, where  ∞ denotes the set of bounded sequences of sets.
Proof.The proof of the theorem follows from Theorem 9 and Lemma 10.