Statistical Convergence of Nets in Riesz Spaces

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x ∧ y = inf{x,y} and x ∨ y = sup{x,y} exist in E, respectively. A Riesz space is called a Dedekind complete if every nonempty bounded from the above set has a supremum (or, equivalently, whenever every nonempty bounded below subset has an inﬁmum). A subset I of a Riesz space E is said to be a solid if, for each x ∈ E and y ∈ I with |x| ≤ |y|, it follows that x ∈ I. A

2 solid vector subspace is called an order ideal. A Riesz space E has 1 the Archimedean property provided that n x ↓ 0 holds in E for each x ∈ E+. In this paper, unless otherwise stated, all Riesz spaces are assumed to be real and Archimedean. We remind the crucial notion of Riesz spaces (cf. [1, 2, 16, 24]).

Deﬁnition 1.1. A net (xα)α∈A in a Riesz space E is called order convergent to x ∈ E if there exists another net (yα)α∈A ↓ 0 (i.e., inf yα = 0 and yα ↓) such that |xα − x|≤ yα holds for all α ∈ A. We refer to the reader for some diﬀerent types of the order convergence and some relations among them to [3].

2 The µ-statistically order convergence

We remind that a map from a field M to [0, ∞] is called finitely n n additive measure whenever µ(∅)=0 and µ(∪i=1Ei)= Pi=1 µ(Ei) for n all finite disjoint sets {Ei}i=1 in M (cf. [15, p.25]). Now, we introduce the notion of measure on directed sets. Definition 2.1. Let A be a directed set and M be a subfield of P(A). Then (1) an order interval [a, b] of A is said to be a finite order interval if it is a finite subset of A; (2) M is called an interval field on A whenever it includes all finite order intervals of A; (3) a finitely additive measure µ : M→ [0, 1] is said to be a directed set measure if M is an interval field and µ satisfies the following facts: µ(I) = 0 for every finite order interval I ∈ M; µ(A) = 1; µ(C) = 0 whenever C ⊆ B and µ(B) = 0 holds for B,C ∈ M. Example 2.2. Consider the directed set A := N and define a measure N 1 µ from 2 to [0, 1] denoted by µ(A) as the Banach limit of k |A ∩ {1, 2,...,k}| for all A ∈ 2N. Then one can see that µ(I) = 0 for 1 1 all finite order interval sets because of k |I ∩ {1, 2,...,k}| ≤ k I → 0. Also, it follows from the properties of Banach limit that µ(N)=1and µ(A ∪ B)= µ(A)+ µ(B) for disjoint sets A and B. Thus, µ is finitely additive, and so, it is a directed set measure. Throughout this paper, the vertical bar of a set will stand for the cardinality of the given set and P(A) is the power set of A. Let give an example of directed set measure for an arbitrary uncountable set.

3 Example 2.3. Let A be an uncountable directed set. Consider a field M consists of countable or co-countable (i.e., the complement of the set is countable) subsets of A. Then M is an interval field. Thus, a map µ from M to [0, 1] defined by µ(C) := 0 if C is a countable set, otherwise µ(C) = 1. Hence, µ is a directed set measure. In this paper, unless otherwise stated, we consider all nets with a directed set measure on interval fields of the power set of the index sets. Moreover, in order to simplify presentation, a directed set measure on an interval field M of the directed set A will be expressed briefly as a measure on the directed set A. Motivated from [14, p.302], we give the following notion.

Definition 2.4. Ifanet(xα)α∈A satisfies a property P for all α except a subset of A with measure zero then we say that (xα)α∈A satisfies the property P for almost all α, and we abbreviate this by a.a.α. Recall that the asymptotic density of a subset K of natural numbers N is defined by 1 δ(K) := lim |{k ≤ n : k ∈ A}| . n→∞ n We refer the reader for an exposition on the asymptotic density of sets in N to [13, 21]. It can be observed that finite subsets and finite order intervals in natural numbers coincide. Thus, we give the following observation. Remark 2.5. It is clear that the asymptotic density of subsets on N satisfies the conditions of a directed set measure when P(N) is con- sidered as an interval field on the directed set N. Thus, it can be seen that the directed set measure is an extension of the asymptotic density.

Remind that a sequence (xn) in a Riesz space E is called statistically monotone decreasing to x ∈ E if there exists a subset K of N such that δ(K) = 1 and (xnk ) is decreasing to x, i.e., (xnk )k ↓ and inf(xnk ) = x (cf. [5, 23]). Now, by using the notions of measure on directed sets and the statistical monotone decreasing which was introduced in [21] for real sequences, we introduce the concept of the statistical convergence of nets in Riesz spaces.

Deﬁnition 2.6. Let E be a Riesz space and (pα)α∈A be a net in E with a measure µ on the index set A. Then (pα)α∈A is said to

4 be µ-statistically decreasing to x ∈ E whenever there exists ∆ ∈ M such that µ(∆) = 1 and (pαδ )δ∈∆ ↓ x. Then it is abbreviated as stµ (pα)α∈A ↓ x. We denote the class of all µ-statistically decreasing nets in a Riesz space E by Estµ↓, and also, the set Estµ↓{0} denotes the class of all µ-statistically decreasing null nets on E. On the other hand, one can observe that µ(∆c) = µ(A − ∆) = 0 whenever µ(∆) = 1 because of µ(A)= µ(∆ ∪ ∆c)= µ(∆) + µ(∆c). We consider Example 2.3 for the following example.

Example 2.7. Let E be a Riesz space and (pα)α∈A be a net in E. Take M and µ from Example 2.3. Thus, if (pα)α∈A ↓ x then stµ (pα)α∈A ↓ x for some x ∈ E. For the general case of Example 2.7, we give the following work whose proof follows directly from the basic deﬁnitions and results.

Proposition 2.8. If (pα)α∈A is an order decreasing null net in a Riesz stµ space then (pα)α∈A ↓ 0. Now, we introduce the crucial notion of this paper.

Deﬁnition 2.9. A net (xα)α∈A in a Riesz space E is said to be µ- statistically order convergent to x ∈ E if there exists a net (pα)α∈A ∈

Estµ↓{0} with µ(∆) = 1 for some ∆ ∈ M such that (pαδ )δ∈∆ ↓ 0 and stµ |xαδ − x|≤ pαδ for every δ ∈ ∆. Then it is abbreviated as xα −−→ x.

stµ It can be seen that the means of xα −−→ x in a Riesz space is stµ equal to say that there exists another sequence (pα)α∈A ↓ 0 such that µ{α ∈ A : |xα − x| pα} = 0. It follows from Remark 2.5 that the notion of statistical convergence of nets coincides with the notion of µ-statistically order convergence in reel line. We denote the set Estµ as the family of all stµ-convergent nets in E, and Estµ {0} is the family of all µ-statistically order null nets in E.

Lemma 2.10. Every µ-statistically decreasing net µ-statistically order converges to its µ-statistically decreasing limit in Riesz spaces.

Remark 2.11. Recall that a net (xα)α∈A in a Riesz space E relatively uniform converges to x ∈ E if there exists u ∈ E+ such that, for N 1 any n ∈ , there is an index αn ∈ A so that |xα − x| ≤ n u for all α ≥ αn (cf. [16, Thm.16.2]). It is well known that the relatively

5 uniform convergence implies the order convergence on Archimedean Riesz spaces (cf. [6, Lem.2.2]). Hence, it follows from Proposition 2.8 and Lemma 2.10 that every decreasing relatively uniform null net is µ-statistically order convergent in Riesz spaces.

3 Main Results

Let µ be a measure on a directed set A. Following from [15, Exer.9. p.27], it is clear that µ(∆∩Σ) = 1forany∆, Σ ⊆ A, µ(∆) = µ(Σ) = 1. So, we begin the section with the following observation.

Proposition 3.1. Assume xα ≤ yα ≤ zα satisﬁes in a Riesz spaces stµ stµ stµ for each index α. Then yα −−→ x whenever xα −−→ x and zα −−→ x.

It can be seen from Proposition 3.1 that if 0 ≤ xα ≤ zα satisﬁes for each index α and (zα)α∈A ∈ Estµ {0} then (xα)α∈A ∈ Estµ {0}. We give a relation between the order and the µ-statistically order convergences in the next result. Theorem 3.2. Every order convergent net is µ-statistically order convergent to its order limit.

Proof. Suppose that a net (xα)α∈A is order convergent to x in a Riesz space E. Then there exists another net (yα)α∈A ↓ 0 such that |xα − x| ≤ yα holds for all α ∈ A. It follows from Proposition 2.8 that stµ stµ (yα)α∈A ↓ 0. So, we obtain the desired result, (xα)α∈A −−→ x. The converse of Theorem 3.2 need not to be true as the following example shows

Example 3.3. Take the sequence (en) of the standard unit vectors in the Riesz space c0 the set of all null sequence in the real numbers. Then (xn) = (e1, 0, e2, 0, e3, · · · ) does not converge in order because it is not order bounded in c0. However, it can be seen that all eventually zero subsequences of (en) is order convergent to zero, and so, they are µ-statistically order convergent to zero. Then it follows Proposition stµ 3.5 that xn −−→ 0. Following from Theorem 3.2 and [16, Thm.23.2], we observe the following result. Corollary 3.4. Every order bounded monotone net in a Dedekind complete Riesz space is µ-statistically order convergent.

6 Proposition 3.5. The stµ-convergence of subnets implies the stµ- convergence of nets.

Proof. Let (xα)α∈A be a net in a Riesz space E. Assume that a subnet

(xαδ )δ∈∆ of (xα)α∈A µ-statistically order converges to x ∈ E. Then there exists a net (pαδ )δ∈∆ ∈ Estµ↓{0} with Σ ⊆ ∆ and µ(Σ) = 1 such p x x p σ A that ( αδσ )σ∈Σ ↓ 0 and | αδσ − |≤ αδσ for each ∈ Σ. Since Σ ⊆ x x and ( αδσ )σ∈Σ is also a subnet of ( α)α∈A, we can obtain the desired result.

Since every order bounded net has an order convergent subnet in atomic KB-spaces, we give the following result by considering Theo- rem 3.2 and proposition 3.5.

Corollary 3.6. If E is an atomic KB-space then every order bounded net is µ-statistically order convergent in E.

The lattice operations are µ-statistically order continuous in the following sense.

stµ stµ stµ Theorem 3.7. If xα −−→ x and wα −−→ w then xα ∨ wα −−→ x ∨ w.

stµ stµ Proof. Assume that xα −−→ x and wα −−→ w hold in a Riesz space E.

So, there are nets (pα)α∈A, (qα)α∈A ∈ Estµ↓{0} with ∆, Σ ∈ M and µ(∆) = µ(Σ) = 1 such that

|xαδ ∨ wασ − x ∨ w|≤ pαδ + qασ for each δ ∈ ∆ and σ ∈ Σ. Take Γ := ∆ ∩ Σ ∈ M. So, we have

µ(Γ) = 1, and also, |xαγ ∨ wδγ − x ∨ w|≤ pαγ + qδγ holds for all γ ∈ Γ. stµ It follows from (pαγ + qδγ )γ∈Γ ↓ 0 that xα ∨ wα −−→ x ∨ w.

stµ stµ Corollary 3.8. If xα −−→ x and wα −−→ w in a Riesz space then

stµ (i) xα ∧ wα −−→ x ∧ w;

stµ (ii) |xα| −−→|x|;

7 + stµ + (iii) xα −−→ x ; − stµ − (iv) xα −−→ x . We continue with several basic results that are motivated by their analogies from Riesz space theory.

Theorem 3.9. Let (xα)α∈A be a net in a Riesz space E. Then the following results hold:

stµ stµ stµ (i) xα −−→ x iﬀ (xα − x) −−→ 0 iﬀ |xα − x| −−→ 0; (ii) the µ-statistically order limit is linear; (iii) the µ-statistically order limit is uniquely determined;

(iv) the positive cone E+ is closed under the µ-statistically order convergence;

stµ stµ (v) xαδ −−→ x for any subnet (xαδ )δ∈∆ of xα −−→ x with µ(∆) = 1. Proof. The properties (i), (ii) and (iii) are straightforward. For (iv), take a non-negative µ-statistically order convergent net stµ + stµ + xα −−→ x in E. Then it follows from Corollary 3.8 that xα = xα −−→ x . Moreover, by applying (iii), we have x = x+. So, we obtain the desired result x ∈ E+. stµ For (v), suppose that xα −−→ x. Then there is a net (pα)α∈A ∈

Estµ↓{0} with ∆ ∈ M and µ(∆) = 1 such that |xαδ − x| ≤ pαδ for stµ each δ ∈ ∆. Thus, it is clear that xαδ −−→ x. However, it should be shown that it is provided for all subnets under our assumption. Thus, take an arbitrary element Σ ∈ M with Σ 6= ∆ and µ(Σ) = 1. We show stµ (xασ )σ∈Σ −−→ x. Consider Γ := ∆ ∩ Σ ∈ M. So, we have µ(Γ) = 1.

Therefore, following from |xαγ − x| ≤ pαγ for each γ ∈ Γ, we get the desired result.

Theorem 3.2 shows that the order convergence implies the µ-statistically order convergence. For the converse of this, we give the following result.

Theorem 3.10. Every monotone µ-statistically order convergent net order converges to its stµ-limit in Riesz spaces.

stµ Proof. We show that xα ↓ and xα −−→ x implies xα ↓ x in any Riesz space E. To see this, choose an arbitrary index α0. Then xα0 − xα ∈

8 stµ E+ for all α ≥ α0. It follows from Theorem 3.9 that xα0 −xα −−→ xα0 − x, and also, xα0 − x ∈ E+. Hence, we have xα0 ≥ x. Then x is a lower bound of (xα)(α∈A) because α0 is arbitrary. Suppose that z is another stµ lower bound of (xα)α∈A. So, we obtain xα − z −−→ x − z. It means that x − z ∈ E+, or equivalent to saying that x ≥ z. Therefore, we get xα ↓ x.

Theorem 3.11. Let x := (xα)α∈A be a net in a Riesz space. If o stµ xX∆ −→ 0 holds for some ∆ ∈ M with µ(∆) = 1 then x −−→ 0. o Proof. Suppose that there exists ∆ ∈ M with µ(∆) = 1 and xX∆ −→ 0 satisﬁes in a Riesz space E for the characteristic function X∆ of ∆. Thus, there is another net (pα)α∈A ↓ 0 such that |xX∆| ≤ pα for all stµ α ∈ A. So, it follows from Proposition 2.8 that (pα)α∈A ↓ 0. Then there exists a subset Σ ∈ M such that µ(Σ) = 1 and (pασ )σ∈Σ ↓ 0. Take Γ := ∆ ∩ Σ. Hence, we have µ(Γ) = 1. Following from |xXΓ|≤ stµ pαγ for each γ ∈ Γ, we obtain xX∆ −−→ 0. Therefore, by applying stµ Proposition 3.5, we obtain (xα)α∈A −−→ 0.

Proposition 3.12. The family of all stµ-convergent nets Estµ is a Riesz space.

Proof. From the properties in Theorem 3.9, Estµ is a vector space. stµ Take an element x := (xα)α∈A in Estµ . Then we have x −−→ z for stµ some z ∈ E. Thus, it follows from Corollary 3.8 that |x| −−→|z|. It means that |x| ∈ Estµ , i.e., Estµ is a Riesz subspace (cf. [1, Thm.1.3 and Thm.1.7]).

Theorem 3.13. The set of all order bounded nets in a Riesz space E is an order ideal in Estµ {0}.

Proof. By the same argument in Proposition 3.12, Estµ {0} is a Riesz space. Assume that |y| ≤ |x| hold for arbitrary x := (xα)α∈A ∈ stµ Estµ {0} and for an order bounded net y := (yα)α∈A. Since x −−→ 0, stµ stµ we have |x| −−→ 0. Then it follows from Proposition 3.1 that |y| −−→ 0, stµ and so, it follows from Theorem 3.9(i) that y −−→ 0. Therefore, we get the desired result, y ∈ Estµ {0}.

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