A note on statistical convergence on time scale
© Seyyidoglu and Tan; licensee Springer 2012
Received: 14 June 2012
Accepted: 11 September 2012
Published: 2 October 2012
In the present paper, we will give some new notions, such as Δ-convergence and Δ-Cauchy, by using the Δ-density and investigate their relations. It is important to say that the results presented in this work generalize some of the results mentioned in the theory of statistical convergence.
1 Introduction and background
In  Fast introduced an extension of the usual concept of sequential limits which he called statistical convergence. In  Schoenberg gave some basic properties of statistical convergence. In  Fridy introduced the concept of a statistically Cauchy sequence and proved that it is equivalent to statistical convergence.
The theory of time scales was introduced by Hilger in his PhD thesis supervised by Auldbach  in 1988. The measure theory on time scales was first constructed by Guseinov , and then further studies were performed by Cabada-Vivero  and Rzezuchowski . In  Deniz-Ufuktepe define Lebesgue-Stieltjes Δ and ▽-measures, and by using these measures, they define an integral adapted to a time scale, specifically Lebesgue-Steltjes Δ-integral.
A time scale is an arbitrary nonempty closed subset of the real numbers ℝ. The time scale is a complete metric space with the usual metric. We assume throughout the paper that a time scale has the topology that it inherits from the real numbers with the standard topology.
In this definition, we put .
Open intervals and half-open intervals etc. are defined accordingly.
The collection of all -measurable sets is a σ-algebra, and the restriction of to , which we denote by , is a countably additive measure on . We call this measure , which is the Carathéodory extension of the set function m associated with the family , the Lebesgue Δ-measure on .
We call a measurable function, if for every open subset of of ℝ.
Theorem 1 (see )
Theorem 2 (see )
It can easily be seen from Theorem 1 that the measure of a subset of ℕ is equal to its cardinality.
2 Δ-density, Δ-convergence, Δ-Cauchy
It is well known that the notions of statistical convergence and statistical Cauchy are closely related to the density of the subset of ℕ. In the present section, first of all, we will define the density of the subset of the time scale. By using this definition, we will define Δ-convergence and Δ-Cauchy for a real valued function defined on the time scale. Then we will show that these notions are equivalent.
Throughout this paper, we consider the time scales which are unbounded from above and have a minimum point.
From the identity , the measurability of A implies the measurability of .
If and , then .
If , then .
If , then and .
If and , then and .
- (vi)If is a mutually disjoint sequence in , then and
- (vii)If in and , then
If A is a measurable set and with , then .
If in , then , .
Every bounded measurable subset of belongs to .
If and , then .
- (ii)Note that . The required inequalities follow from the following inequalities:
- (iii)It is clear that is measurable. The Δ-density of is obtained from the following equalities:
- (iv)Since A is measurable, so is , namely is measurable. On the other hand, and
Since A and B are measurable, so are . The statement can be easily shown by considering .
- (vi)Since the Δ-density of for each subset exists, one can write
The proof is similar to that of the previous proof.
It can be easily seen from (i).
Considering for and (vii), one can obtain . And can be obtained from (viii).
- (x)Let A be a bounded set. For a sufficiently large , we can write . Then one has
(i) and (vii) yield . This completes the proof. □
It is clear that the family is a ring of subsets of . According to (iv), the Δ-density of the complement of a subset whose Δ-density is 0 is equal to 1, is not closed under the operation complement. So, it is not an algebra. Note that the Δ-density of a subset of ℕ is equal to its natural density.
does not define a measure.
Definition 1 (Δ-convergence)
The function is Δ-convergent to the number L provided that for each , there exists such that and holds for all .
We will use notation .
Definition 2 (Δ-Cauchy)
The function is Δ-Cauchy provided that for each , there exists and such that and holds for all .
Proposition 1 Let be a measurable function. Δ- if and only if, for each , .
Proof Let and be given. In this case, there exists a subset such that and holds for all . Since , we obtain . Hence, we get .
Another case of the proof is straightforward. □
Proposition 2 Let be a measurable function. f is Δ-Cauchy if and only if, for each , there exists such that .
Since , the density of the subset in the time scale is zero. This implies that . So, for each and for all , one has , and as a corollary, we get .
Proposition 3 The Δ-limit of a function is unique.
Thus, . □
Proposition 5 If with , then .
Proof Let . In this case, for a given , we can find a such that holds for every . The set is measurable, and from Lemma 1(iv) and (x), one has . By the definition of Δ-convergence, we get . □
f is Δ-convergent,
f is Δ-Cauchy,
There exists a measurable and convergent function such that for Δ-a.a. t.
- (ii)⇒ (iii): We can choose an element . We can define an interval which contains for Δ-a.a. t. By the same method, we can choose an element and define an interval which contains for Δ-a.a. t. We can write
It is clear that g is a measurable function and . Indeed, for , either or . In this case, holds.
- (iii)⇒ (i): Let for Δ-a.a. t and . For a given , we have
Since , the second set that appears on the right-hand side of the above inclusion relation is bounded, and thus . In addition, for Δ-a.a. t yields . In conclusion, , namely . □
The editor and referee(s) remark that the results obtained in this paper and many other characterizations (not included here) have already been presented independently with a similar title by C. Turan and O. Duman at the AMAT 2012 - International Conference on Applied Mathematics and Approximation Theory, May 17-20, 2012, Ankara-Turkey (http://amat2012.etu.edu.tr) and submitted to the Springer Proceeding.
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