Abstract
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.
MSC:34N05.
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1 Introduction and background
In [1] Fast introduced an extension of the usual concept of sequential limits which he called statistical convergence. In [2] Schoenberg gave some basic properties of statistical convergence. In [3] 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 [4] in 1988. The measure theory on time scales was first constructed by Guseinov [5], and then further studies were performed by Cabada-Vivero [6] and Rzezuchowski [7]. In [8] 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.
For , we define the forward jump operator by
In this definition, we put .
For with , we define the interval in by
Open intervals and half-open intervals etc. are defined accordingly.
Let be a time scale. Denote by the family of all left-closed and right-open intervals of of the form with and . The interval is understood as an empty set. is a semiring of subsets of . Obviously, the set function defined by is a countably additive measure. An outer measure generated by m is defined by
If there is no sequence of such that , then we let . We define the family of all -measurable subsets of , i.e.,
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 [5])
For each in , the single point set is Δ-measurable, and its Δ-measure is given by
Theorem 2 (see [5])
If and , then
If and , then
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.
Let A be a Δ-measurable subset of and let . Δ-density of A in (or briefly Δ-density of A) is defined by
(if this limit exists), where
and .
From the identity , the measurability of A implies the measurability of .
If is a function such that satisfies the property P for all t except a set of Δ-density zero, then we say that satisfies P for ‘Δ-almost all t’, and we abbreviate this by ‘Δ-a.a. t’. Let
Lemma 1
-
(i)
If and , then .
-
(ii)
If , then .
-
(iii)
and .
-
(iv)
If , then and .
-
(v)
If and , then and .
-
(vi)
If is a mutually disjoint sequence in , then and
-
(vii)
If in and , then
-
(viii)
If A is a measurable set and with , then .
-
(ix)
If in , then , .
-
(x)
Every bounded measurable subset of belongs to .
-
(xi)
If and , then .
Proof (i) Let and . Clearly, , and since is a measure function, one has . Thus, we have
-
(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
imply that the required statement holds as .
-
(v)
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
-
(vii)
The proof is similar to that of the previous proof.
-
(viii)
It can be easily seen from (i).
-
(ix)
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
which implies that .
-
(xi)
(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.
Example 1 Let , l and r be arbitrary two positive real numbers. Let also , where . According to Lemma 1(x), each is bounded and so . In addition, let be defined as in (2.1), we have
, and hence
Note that since
does not define a measure.
Example 2 Let . The Δ-density of in ℕ is given by
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 .
Example 3 Let be irrational numbers and be rational numbers in . Let us consider the function defined as follows:
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.
Proof Let and . Let be given. Then there exist subsets such that for every with and for every with , where and . From Lemma 1(iv), we have . Thus, for every , one has
Thus, . □
Proposition 4 If with and , then the following statements hold:
-
(i)
,
-
(ii)
().
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 . □
Theorem 3 Let be a measurable function. The following statements are equivalent:
-
(i)
f is Δ-convergent,
-
(ii)
f is Δ-Cauchy,
-
(iii)
There exists a measurable and convergent function such that for Δ-a.a. t.
Proof (i) ⇒ (ii): Let and be given. Then holds for Δ-a.a. t. We can choose such that holds. So,
This shows that f satisfies the property of Δ-Cauchy.
-
(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
Since f is a measurable function, the two terms that appear on the right-hand side of the last equality are also measurable. By using Lemma 1(vii), we obtain
So, is in the closed interval for Δ-a.a. t. It is clear that the length of the interval is less than or equal to 1. Now we can choose with containing for Δ-a.a. t. Undoubtedly, the closed interval contains for Δ-a.a. t and the length of the interval is less than or equal to . With the same procedure, for each m, we can obtain a sequence of closed intervals such that and the length of each interval is less than or equal to . Moreover, for Δ-a.a. t. From the properties of the intersection of closed intervals, there exists a real number λ such that . Since the Δ-density of the set on which is equal to zero, we can find an increasing sequence in such that
Here . Let us consider the function defined as follows:
It is clear that g is a measurable function and . Indeed, for , either or . In this case, holds.
Finally, we shall show that for Δ-a.a. t. For this purpose, consider
. Thus, from (2.2), we get
which yields , that is, for Δ-a.a. t.
-
(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 . □
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Acknowledgements
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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MSS carried out the studies on Functional Analysis, participated theory of statistical convergence and their relations. NOT participated in the measure theory studies and performed the relation convergence with measure and drafted the manuscript.
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Seyyidoglu, M.S., Tan, N.Ö. A note on statistical convergence on time scale. J Inequal Appl 2012, 219 (2012). https://doi.org/10.1186/1029-242X-2012-219
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DOI: https://doi.org/10.1186/1029-242X-2012-219