R E S E A R C H
Open Access
A note on statistical convergence on time
scale
M Seyyit Seyyidoglu and N Özkan Tan
**Correspondence: nozkan.tan@usak.edu.tr Faculty of Sciences and Arts, Department of Mathematics, Usak University, 1 Eylul Campus, Usak, 64200, Turkey
Abstract
In the present paper, we will give some new notions, such as
-convergence and
-Cauchy, by using the
-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
Keywords: time scale; Lebesgue
-measure; 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 Auld-bach [] in . 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-Stieltjesand-measures, and by using these measures, they define an integral adapted to a time scale, specifically Lebesgue-Steltjes -integral.
A time scaleTis an arbitrary nonempty closed subset of the real numbersR. The time scaleTis a complete metric space with the usual metric. We assume throughout the paper that a time scaleThas the topology that it inherits from the real numbers with the standard topology.
Fort∈T, we define theforward jump operatorσ:T→Tby
σ(t) :=inf{s∈T:s>t}.
In this definition, we putinf∅=supT.
Fora,b∈Twitha≤b, we define the interval [a,b] inTby
[a,b] ={t∈T:a≤t≤b}.
Open intervals and half-open intervalsetc.are defined accordingly.
LetTbe a time scale. Denote bySthe family of all left-closed and right-open intervals ofTof the form [a,b) ={t∈T:a≤t<b}witha,b∈Tanda≤b. The interval [a,a) is
understood as an empty set.S is a semiring of subsets ofT. Obviously, the set function
m:S→[,∞] defined bym([a,b)) =b–ais a countably additive measure. An outer measurem∗:P(T)→[,∞] generated bymis defined by
m∗(A) =inf
∞
n=
m(An) : (An) is a sequence ofSwithA⊂ ∞
n=
An
.
If there is no sequence (An) ofSsuch thatA⊂
∞
n=An, then we letm∗(A) =∞. We define the familyM(m∗) of allm∗-measurable subsets ofT,i.e.,
Mm∗= E⊂T:m∗(A) =m∗(A∩E) +m∗A∩Ecfor allA⊂T.
The collectionM(m∗) of allm∗-measurable sets is aσ-algebra, and the restriction ofm∗
toM(m∗), which we denote byμ, is a countably additive measure onM(m∗). We call this measureμ, which is the Carathéodory extension of the set functionmassociated with the familyS, the Lebesgue-measure onT.
We callf :T→Ra measurable function, iff–(O)∈M(m∗) for every open subset ofO
ofR.
Theorem (see []) For each tinT–{maxT},the single point set{t}is-measurable,
and its-measure is given by
μ{t}
=σ(t) –t.
Theorem (see []) If a,b∈Tand a≤b,then
μ
[a,b)=b–a, μ
(a,b)=b–σ(a).
If a,b∈T–{maxT}and a≤b,then
μ(a,b]=σ(b) –σ(a), μ[a,b]=σ(b) –a.
It can easily be seen from Theorem that the measure of a subset ofNis equal to its cardinality.
2
-density,
-convergence,
-Cauchy
It is well known that the notions of statistical convergence and statistical Cauchy are closely related to the density of the subset ofN. 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 -convergence and-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.
LetAbe a-measurable subset ofTand leta=minT.-density ofAinT(or briefly -density ofA) is defined by
δ(A) = lim
t→∞
(if this limit exists), where
A(t) ={s∈A:s≤t} (.)
andt∈T.
From the identity A(t) =A∩[a,t], the measurability of A implies the measurability ofA(t).
Iff :T→Ris a function such thatf(t) satisfies the propertyPfor alltexcept a set of -density zero, then we say thatf(t) satisfiesPfor ‘-almost allt’, and we abbreviate this by ‘-a.a.t’. Let
Md:={A∈T:-density ofAexists inT}, M
d:= A∈T:δ(A) =
.
Lemma
(i) IfA,B∈MdandA⊂B,thenδ(A)≤δ(B).
(ii) IfA∈Md,then≤δ(A)≤.
(iii) T∈Mdandδ(T) = .
(iv) IfA∈Md,thenAc∈Mdandδ(A) +δ(Ac) = .
(v) IfA,B∈MdandA⊂B,thenB–A∈Mdandδ(B–A) =δ(B) –δ(A).
(vi) IfA,A, . . . ,Anis a mutually disjoint sequence inMd,then
n
k=Ak∈Mdand
δ
n
k=
Ak
= n
k=
δ(Ak).
(vii) IfA,A, . . . ,AninMdand
n
k=Ak∈Md,then
δ
n
k=
Ak
≤
n
k=
δ(Ak).
(viii) IfAis a measurable set andB∈M
dwithA⊂B,thenA∈Md.
(ix) IfA,A, . . . ,AninMd,then
n
k=Ak,
n
k=Ak∈Md.
(x) Every bounded measurable subset ofTbelongs toMd. (xi) IfA∈M
dandB∈Md,thenδ(A∪B) =δ(B).
Proof (i) LetA,B∈MdandA⊂B. Clearly,A(t)⊂B(t), and sinceμis a measure func-tion, one hasμ(A)≤μ(B). Thus, we have
δ(A) = lim
t→∞
μ(A(t)) σ(t) –a ≤tlim→∞
μ(B(t))
σ(t) –a =δ(B).
(ii) Note thatA(t)⊂[a,t]. The required inequalities follow from the following inequali-ties:
≤μ(A(t)) σ(t) –a ≤
μ([a,t]) σ(t) –a =
σ(t) –a
(iii) It is clear thatTis measurable. The-density ofTis obtained from the following equalities:
δ(T) = lim
t→∞
μ(T(t)) σ(t) –a =tlim→∞
μ([a,t]) σ(t) –a = .
(iv) SinceAis measurable, so isAc, namelyAc(t) ={s≤t:s∈Ac}is measurable. On the other hand,A(t)∪Ac(t) = [a,t] and
=μ([a,t]) σ(t) –a =
μ(A(t)) σ(t) –a +
μ(Ac(t)) σ(t) –a
imply that the required statement holds ast→ ∞.
(v) SinceAandBare measurable, so areB–A. The statement can be easily shown by consideringA(t)∪(B–A)(t)∪Bc(t) = [a,t].
(vi) Since the-density of for each subsetAkexists, one can write
n
k=
δ(Ak) = n
k=
lim
t→∞
μ(Ak(t)) σ(t) –a
= lim
t→∞ n
k=
μ(Ak(t)) σ(t) –a
= lim
t→∞
μ(nk=Ak(t)) σ(t) –a
= lim
t→∞
μ((nk=Ak)(t)) σ(t) –a
=δ
n
k=
Ak
.
(vii) The proof is similar to that of the previous proof. (viii) It can be easily seen from (i).
(ix) Consideringδ(Ak) = fork= , , . . . ,nand (vii), one can obtain
n
k=Ak∈Md. Andnk=Ak∈Mdcan be obtained from (viii).
(x) LetAbe a bounded set. For a sufficiently largeM∈T, we can writeA⊂[a,M]. Then one has
≤μ(A(t)) σ(t) –a ≤
μ([a,M])
σ(t) –a → (t→ ∞),
which implies thatδ(A) = .
(xi) (i) and (vii) yieldδ(B)≤δ(A∪B)≤δ(A) +δ(B) =δ(B). This completes the
proof.
It is clear that the familyM
Example LetT= [,∞),landrbe arbitrary two positive real numbers. Let alsoA=
∞
n=An, whereAn= [nl+nr, (n+ )l+nr]. According to Lemma (x), eachAnis bounded and soδ(An) = . In addition, letA(t) be defined as in (.), we have
μA(t)=
⎧ ⎨ ⎩
t–nr, nl+nr≤t≤(n+ )l+nr,
(n+ )l, (n+ )l+nr≤t≤(n+ )l+ (n+ )r
(n= , , , . . .), and hence
δ(A) = lim
t→∞
μ(A(t)) σ(t) –a =tlim→∞
μ(A(t))
t– =
l l+r.
Note that since
l
l+r=δ(A) >
∞
n=
δ(An) = ,
δdoes not define a measure.
Example LetT=N. The-density ofA={, , , . . .}inNis given by
δ(A) = lim
t→∞
μ(A(t)) σ(t) –a =tlim→∞
[t/]
t =
.
Definition (-convergence) The functionf :T→Ris-convergent to the numberL
provided that for eachε> , there existsKε⊂Tsuch thatδ(Kε) = and|f(t) –L|<ε holds for allt∈Kε.
We will use notation-limt→∞f(t) =L.
Definition (-Cauchy) The functionf :T→Ris-Cauchy provided that for each ε> , there existsKε⊂Tandt∈Tsuch thatδ(Kε) = and|f(t) –f(t)|<εholds for all
t∈Kε.
Proposition Let f :T→Rbe a measurable function.-limt→∞f(t) =L if and only if,
for eachε> ,δ({t∈T:|f(t) –L| ≥ε}) = .
Proof Let-limt→∞f(t) =Landε> be given. In this case, there exists a subsetKε⊂T such thatδ(Kε) = and|f(t) –L|<εholds for allt∈Kε. SinceKε⊂ {t∈T:|f(t) –L|<ε}, we obtainδ({t∈T:|f(t) –L|<ε}) = . Hence, we getδ({t∈T:|f(t) –L| ≥ε}) = .
Another case of the proof is straightforward.
Proposition Let f :T→Rbe a measurable function.f is-Cauchy if and only if,for eachε> ,there exists t∈Tsuch thatδ({t∈T:|f(t) –f(t)| ≥ε}) = .
Example LetI[,∞)be irrational numbers andQ[,∞)be rational numbers in [,∞). Let
us consider the functionf:T= [,∞)→Rdefined as follows:
f(t) =
⎧ ⎨ ⎩
, t∈Q[,∞),
Sinceμ(Q[,∞)) = , the density of the subset Q[,∞) in the time scaleTis zero. This
implies thatδ(I[,∞)) = . So, for eachε> and for allt∈I[,∞), one has =|f(t) – |<ε,
and as a corollary, we get-limt→∞f(t) = .
Proposition The-limit of a function f :T→Ris unique.
Proof Let-limt→∞f(t) =Land-limt→∞f(t) =L. Letε> be given. Then there exist
subsetsK,K⊂Tsuch that for everyt∈Kwith|f(t) –L|<ε/ and for everyt∈Kwith
|f(t) –L|<ε/, whereδ(K) = andδ(K) = . From Lemma (iv), we haveK∩K=∅.
Thus, for everyt∈K∩K, one has
|L–L| ≤f(t) –L+f(t) –L<ε.
Thus,L=L.
Proposition If f,g:T→Rwith-limt→∞f(t) =L and-limt→∞g(t) =L,then the
following statements hold:
(i) -limt→∞(f(t) +g(t)) =L+L,
(ii) -limt→∞(cf(t)) =cL(c∈R).
Proposition If f :T→Rwithlimt→∞f(t) =L,then-limt→∞f(t) =L.
Proof Letlimt→∞f(t) =L. In this case, for a givenε> , we can find at∈Tsuch that
|f(t) –L|<εholds for everyt>t. The setKε={t∈T:t>t}is measurable, and from
Lemma (iv) and (x), one has δ(Kε) = . By the definition of-convergence, we get
-limt→∞f(t) =L.
Theorem Let f :T→Rbe a measurable function.The following statements are equiv-alent:
(i) f is-convergent, (ii) f is-Cauchy,
(iii) There exists a measurable and convergent functiong:T→Rsuch thatf(t) =g(t)
for-a.a.t.
Proof (i)⇒(ii): Let-limt→∞f(t) =Landε> be given. Then|f(t) –L|<ε/ holds for -a.a.t. We can chooset∈Tsuch that|f(t) –L|<ε/ holds. So,
f(t) –f(t)≤f(t) –L+f(t) –L<ε, -a.a.t.
This shows thatf satisfies the property of-Cauchy.
(ii) ⇒ (iii): We can choose an elementt∈T. We can define an interval I= [f(t) –
,f(t) + ] which containsf(t) for-a.a.t. By the same method, we can choose an element
t∗∈Tand define an intervalI= [f(t∗) – /,f(t∗) + /] which containsf(t) for-a.a.t. We can write
Sincef is a measurable function, the two terms that appear on the right-hand side of the last equality are also measurable. By using Lemma (vii), we obtain
δ t∈T:f(t) /∈I∩I≤δ t∈T:f(t) /∈I+δ t∈T:f(t) /∈I
= .
So,f(t) is in the closed intervalI=I∩Ifor-a.a.t. It is clear that the length of the
intervalIis less than or equal to . Now we can chooset∈TwithI= [f(t) – /,f(t) +
/] containingf(t) for-a.a.t. Undoubtedly, the closed intervalI=I∩Icontains
f(t) for-a.a.t and the length of the intervalI is less than or equal to /. With the
same procedure, for eachm, we can obtain a sequence of closed intervals (Im)∞m=such that
Im+⊂Imand the length of each intervalImis less than or equal to –m. Moreover,f(t)∈Im for-a.a.t. From the properties of the intersection of closed intervals, there exists a real number λsuch that∞m=Im={λ}. Since the-density of the set on whichf(t) /∈Im is equal to zero, we can find an increasing sequence (Tm)∞m=inTsuch that
μ({t≤s:f(t) /∈Im}) σ(s) –a <
m, s>Tm. (.)
Herea=minT. Let us consider the functiong:T→Rdefined as follows:
g(t) =
⎧ ⎨ ⎩
λ, Tm<t≤Tm+andf(t) /∈Im,
f(t), otherwise.
It is clear thatg is a measurable function andlimt→∞g(t) =λ. Indeed, fort>Tm, either
g(t) =λorg(t) =f(t)∈Im. In this case,|g(t) –λ| ≤–mholds.
Finally, we shall show thatf(t) =g(t) for-a.a.t. For this purpose, consider
t≤s:f(t)=g(t)⊂ t≤s:f(t) /∈Im
,
Tm<s≤Tm+. Thus, from (.), we get
μ({t≤s:f(t)=g(t)}) σ(s) –a ≤
μ({t≤s:f(t) /∈Im}) σ(s) –a <
m
which yieldsδ({t∈T:f(t)=g(t)}) = , that is,f(t) =g(t) for-a.a.t.
(iii)⇒(i): Letf(t) =g(t)for-a.a.tandlimt→∞g(t) =L. For a givenε> , we have
t∈T:f(t) –L≥ε⊂ t∈T:f(t)=g(t)∪ t∈T:g(t) –L≥ε.
Sincelimt→∞g(t) =L, the second set that appears on the right-hand side of the above inclusion relation is bounded, and thusδ({t∈T:|g(t) –L| ≥ε}) = . In addition,f(t) =
g(t) for-a.a.tyieldsδ({t∈T:f(t)=g(t)}) = . In conclusion,δ({t∈T:|f(t) –L| ≥
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
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.
Received: 14 June 2012 Accepted: 11 September 2012 Published: 2 October 2012 References
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doi:10.1186/1029-242X-2012-219