# Notion of New Structure of Uncertain Sequences UsingΔ-Spaces

Authors: Mashael M. AlBaidani
Journal: Journal of Mathematics
 (2022)

---

## Abstract

The object of this paper is to introduce a new approach ofΔ-operator on the theory of uncertainty using sequence spaces. Also, some structural properties will be studied. Moreover, inclusion relations concerning the newly constructed spaces of this paper will be taken care of.


---

## Body

## 1. Introduction, Background, and Preliminaries

It is a well-known fact that the classical measure, observing nonnegativity and countable additivity, plays a vital role in Science, Mathematics, Engineering, and in particular real life as well. However, the measure employed in real life has no countable additivity in many cases. To taste this structure, researchers proposed different measures like fuzzy measures brought introduced in [1] in 1974. There are many types of uncertainties in real life, such as randomness, fuzziness, and uncertainty which consists of both randomness and fuzziness. To measure fuzzy events, in 1978, Zadeh [2] introduced a possibility measure. A probability measure is used to describe a random event. However, the possibility measure has no self-duality. Thus, B. Liu and Y. Liu [3], in 2002, proposed a self-dual measure, the credibility measure. An axiomatic foundation for credibility theory was given in Liu [4]. Since then, credibility theory has been developed uninterruptedly (see the survey in [5]). In [6], Kwakernaak introduced a fuzzy random variable to describe the phenomenon that fuzziness and randomness simultaneously appear in a system. A hybrid variable was introduced by Liu [7] as a measurable function from a chance space to the set of real numbers. Therefore, a concept of the chance measure was introduced by Li and Liu [8]. Furthermore, some convergence results were discussed (see [9]).It is obvious that the classical measure, probability measure, credibility measure, and chance measure proposed by Li and Liu [8] are all special cases of uncertain measures. But possibility measure is not an uncertain measure. Thus, the properties of uncertain measure are also applicable to classical measure, probability measure, credibility measure, and chance measure. Since sequence convergence plays a very important role in the fundamental theory of mathematics, there are many convergence concepts in classical measure theory, probability theory, credibility theory, and chance theory, and the relationships between them are discussed.LetΓ be a nonempty set and L a σ-algebra over Γ. Each element ν∈L is called an event. To measure an uncertain event, uncertain measure was introduced as a set function S satisfying the following axioms:Axiom 1 (normality):MΓ=1Axiom 2 (monotonicity):Mν1≤Mν2 whenever ν1≤ν2Axiom 3 (self-duality):Mν+Mνc=1 for any event νAxiom 4 (countable subadditivity): for every countable sequence of eventsμi, we have
(1)M∪i=1∞μi≤∑i=1∞Mμi.Since uncertainty phenomena exist widely in real life, the concept of uncertain variable was introduced by Liu [10] as a measurable function from an uncertainty space Γ,M,L to the set of real numbers.As in [10, 11], we have following definitions.Definition 1.
For an uncertain variable, the expected value operator is defined as(2)Eς=∫0+∞Mς≥μdμ−∫−∞0Mς≤μdμ,
only if one of the integrals exists.
We call the functionφρ to uncertainty distribution of the uncertain variable ς if
(3)φρ=Mτ∈Γςτ≤ρ.Definition 2.
An uncertain sequenceςj is said to be convergent almost surely (a.s.) to an uncertain variable ς if there exists an event ν with Mν=1 such that
(4)limj⟶∞ςjκ−ςκ=0,
for everyκ∈ν.Definition 3.
An uncertain sequenceς is said to be convergent in measure to an uncertain variable ς if
(5)limj⟶∞Mςjκ−ςκ≥ε=0,
for everyε>0.Definition 4.
Letς,ς1,ς2,⋯ be the uncertainty distributions of uncertain variables. Then, the sequence ςj is called convergence in mean to ς if
(6)limj⟶∞Eςjκ−ςκ=0.Definition 5.
Letϑ,ϑ1,ϑ2,⋯ be uncertain variables with finite expected values ς,ς1,ς2,⋯, respectively. Then, the sequence ςj is called convergence in distribution to ς if ϑn⟶ϑ at any continuous point of ϑ.Definition 6.
Letϑ=κr be a sequence of natural numbers with κ0=0, 0<κr<κr+1 and hr=κr−κr−1⟶∞ as r⟶∞. Then, ϑ is called a lacunary sequence.
The intervals computed byϑ are abbreviated by Jr=κr−1,κr, and the quotient κr/κr−1 will be symbolized by qr (see [12]).Definition 7 see [13].
A sequence spaceE is said to be solid or normal if ςj∈E implies βjvj∈E for all sequence of scalars βj with bj≤1 for all j∈ℕ.Lemma 8.
If a sequence space is solid, then it is monotone.
In [14], the spaces TΔ were studied and are defined as follows:
(7)TΔ=v=vi∈Λ:Δvi∈T,where T∈ℓ∞,c,C0 and Δvi=vi−vi−1 and were studied further in [15–17] and many others.Next, for integers≥0, the author in [18] had studied the following space:
(8)ΔsT=v=vk:Δsv∈T,forT=ℓ∞,candC0,where Δsvi=Δs−1vi−Δs−1vi+1 for all i∈ℕ.Also, letg=gj be any fixed sequence of nonzero complex numbers, and then, as in [19], we have
(9)ΔgsT=v=vj∈Λ:Δgsvj∈T,where
(10)Δgsvj=Δgs−1vj−Δgs−1vj+1=∑μ=0s−1μsμgj+μvj+μ∀j∈ℕ.It is shown that the spaceΔgsT is Banach under the norm
(11)vΔ=∑i=1sgivi+Δgsv∞.Many interesting structures towards this space can be searched in [7, 13, 20–23] and many others.Inspired by this, in this paper, we interact withΔ-operator to uncertain sequences with the combination of lacunary sequences and syntheses some results in this direction.


## 2. Main Results

Using the concept ofΔ-operator, we deal in this section to introduce some new kind of sequences of uncertain variables.Following the authors [9, 11, 13, 17, 24–27], we introduce the following new spaces:
(12)NθU,Δgs0=ς=ςi:limj⟶∞1hj∑i∈IjΔgsςjv=0,NθU,Δgsc=ς=ςi:limj⟶∞1hj∑i∈IjΔgsςjv−σv=0,NθU,Δgs∞=ς=ςi:limj⟶∞1hj∑i∈IjΔgsςjv<∞,where σv∈Γ,L,S.Theorem 9.
The setsNθU,Δgs0, NθU,Δgsc, and NθU,Δgs∞ of complex certain sequences are linear.Proof.
In order to establish the result, we only consider the case ofNθU,Δgs0 and the rest will follow on similar lines. So, let uj,wj∈NθU,Δgs0, and then,
(13)limj⟶∞1hj∑i∈IjΔgsujv=0,limj⟶∞1hj∑i∈IjΔgswjv=0.
Now, for anya,b∈ℂ, we have
(14)limj⟶∞1hj∑i∈IjΔgsaujv+bujv=limj⟶∞1hj∑i∈IjaΔgsujv+bΔgsujv≤alimj⟶∞1hj∑i∈IjΔgsujv+blimj⟶∞1hj∑i∈IjΔgswjv⟶0,asj⟶∞.
Consequently,aujv+bujv∈NθU,Δgs0 and the result follows.We state the following result without proof.Theorem 10.
The setsNθU,Δgs0, NθU,Δgsc, and NθU,Δgs∞ of complex certain sequences are normed linear spaces with norm
(15)ςvΔgs=∑k=1ngkςkv+supj1hj∑i∈IjΔgsςjv.Theorem 11.
ForT=0,c,∞ and s≥1, we have
(16)NθU,Δgs−1T⊂NθU,ΔgsT.The inclusions are sharp.Proof.
In order to establish the result, we only consider the case ofNθU,Δgs0 and the rest will follow on similar lines. So, let ςj∈NθU,Δgs−10, and then, we see
(17)limj⟶∞1hj∑i∈IjΔgs−1ςiv=0.
We can write(18)1hj∑i∈IjΔgsςiv=1hj∑i∈IjΔgs−1ςiv−Δgs−1ςi+1v≤1hj∑i∈IjΔgs−1ςiv−1hj∑i∈IjΔgs−1ςiv.
Using (1) and letting j⟶∞ in the above inequality, we see
(19)1hj∑i∈IjΔgsςiv=0.
Consequently,ςj∈NθU,Δgs0.
Now to prove the sharpness, we consider the lacunary sequenceθ=2i and choose the sequence of uncertain variables as ςi=is−1 and gi=1 for all i∈ℕ. Then, clearly for all i∈ℕ, we see
(20)Δgsςi=0,Δgsςi=∑r=0s−1−1rs−1rςi+r.
Hence,ςi∈NθU,Δgs0 but not NθU,Δgs−10, as desired.Theorem 12.
ForT=0,c,∞ and s≥1, the spaces of uncertain sequences NθU,ΔgsT are not symmetric in general.Proof.
In order to establish the result, we only consider the case ofNθU,Δgs0 and the rest will follow on similar lines by choosing s=2 and gi=1 for all i∈ℕ and choose a lacunary sequence θ=2i. Choose the uncertainty space Γ,L,M to be τ1,τ2,⋯ with the power set and choose any event ν∈L such that
(21)Mν=supτi∈νi2i+1,ifsupτi∈νi2i+1<0.5,1−supτi∈νci2i+1,ifsupτi∈νci2i+1<0.5,0.5,ifelsewhere.
Further, we define(22)ςiτj=i,ifi=j,0,ifelsewhere.
Then, it is easy to see thatςj for j∈Ij and j=1,2,⋯ is in NθU,Δgs0. We now define the rearrangement of ςj as ωi defined by
(23)ωi=τ=ς1,ς4,ς9,ς2,ς10,⋯,which is obviously not in NθU,Δgs0. This shows that NθU,Δgs0 is not symmetric in general, as desired.In a similar way, we state the following theorem without proof.Theorem 13.
ForT=0,c,∞ and s≥1, the spaces of uncertain sequences NθU,ΔgsT are not monotone in general.


## 3. Lacunary Convergence with respect to Mean

In this section, we introduce the convergence notion ofΔ-operator of uncertain sequences and compute some relation concerning them.In this regard, we have following definitions.Definition 14.
An uncertain sequenceςj is said to be lacunary strongly convergent almost surely to ς w.r.t. difference sequence if for ε>0 there exists an event ν with Mν=1 such that
(24)limj⟶∞1hj∑l∈IjΔgsςlκ−σκ=0,
for everyκ∈ν.Definition 15.
An uncertain sequenceςj is said to be lacunary strongly convergent in measure to ς if
(25)limj⟶∞Mκ∈Γ:1hj∑l∈IjΔgsςlκ−σκ>ε=0,
for everyε>0.Definition 16.
Letς,ς1,ς2,⋯ be the uncertainty distributions of uncertain variables. Then, the sequence ςj is called convergence in mean to ς if
(26)limj⟶∞E1hj∑l∈IjΔgsςlκ−σκ=0.Definition 17.
Letϑ1,ϑ2,ϑ3,⋯ be uncertain variables with finite expected values ς,ς1,ς2,⋯, respectively. Then, the sequence ςj is called lacunary strong convergent in distribution to ς w.r.t. difference sequence if
(27)limj⟶∞1hj∑l∈IjΔgsϑlλ−σλ=0,
for all complexλ at which ϑλ is continuous.Definition 18.
Letϑ1,ϑ2,ϑ3,⋯ be uncertain variables with finite expected values ς,ς1,ς2,⋯, respectively. Then, the sequence ςj is called lacunary strong convergent in distribution to ς w.r.t. difference sequence if
(28)limj⟶∞1hj∑l∈IjΔgsϑlλ−σλ=0,
for all complexλ at which ϑλ is continuous.Definition 19.
Letϑi be uncertain sequence and is said to be convergent uniformly almost surely to ς if there exists a sequence of events Ej, MEj⟶0 such that ϑi converges uniformly to ς in Γ−Ej for any fixed j∈ℕ.Theorem 20.
If the uncertain sequenceςi is lacunary strongly convergent in mean to ς w.r.t. difference sequence, then ςi lacunary strongly converges in measure to ς.Proof.
Givenε>0, we have by Markov’s inequality that
(29)limj⟶∞Mκ∈Γ:1hj∑i∈IjΔgsςiκ−σκ>ε≤limj⟶∞E1/hj∑i∈IjΔgsςiκ−σκε⟶0fori∈Ij.
Thus,ςi lacunary strongly converges in measure to L w.r.t. difference sequence.
To prove the converse may not be true, we choose uncertainty spaceΓ,L,M to be τ1,τ2,⋯ with the power set and choose any event ν∈L such that
(30)Mν=supτi∈ν1i,ifsupτi∈ν1i<0.5,1−supτi∈νc1i,ifsupτi∈νc1i<0.5,0.5,ifelsewhere.
Also, set the uncertain variables as(31)ςiτr=i,ifr=i,0,ifelsewhere,
for alli∈Ij and ς≡0. Now for ε>, we see
(32)limj⟶∞Mκ∈Γ:1hj∑l∈IjΔgsςlκ−σκ>ε=limj⟶∞Mκ∈Γ:1hj∑l∈IjΔgsςlκ>ε=limj⟶∞Mκi=limj⟶∞1i⟶0asi∈Ij.
The sequenceςr lacunary strongly converges in measure to ς. However, for each r∈Ij, we have the uncertainty distribution of uncertain variable ςr−ς=ςr that is
(33)ϑrτ=0,ifτ<0,1−1r,if0≤τ<r,1,ifelsewhere,(34)E1hj∑r∈IjΔgsςrκ−σκ=∫0+∞Mςτdτ−∫−∞0Mς≤τdτ=∫0r1−1−1rdτ=1.
Consequently,ςrτ does not converse in mean to ςτ w.r.t. difference sequence.



---

*Source: 2615772-2022-09-20.xml*