Lacunary ideal convergence in measure for sequences of fuzzy valued functions

Abstract

We investigate the concepts of pointwise and uniform $I_{θ}$ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space $(X, M, μ)$ . We also introduce the concept of $I_{θ}$ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

Keywords

Pointwise convergence uniformly convergence ideal convergence lacunary convergence fuzzy-valued function

1 Introduction and background

Fuzzy set (FS) theory has been demonstrated to be a beneficial tool to explain situations in which the data are imprecise or vague [1]. FSs have been extensively used in several disciplines and technologies. In classical set theory, the membership of elements in a set is evaluated in binary terms according to a bivalent condition, an element either belongs or does not belong to the set. In contrast, FS theory permits the gradual assessment of the membership of elements in a set; this is explained with the help of a membership function valued in the real unit interval [0, 1]. The FS theory can be utilized in a extensive range of domains in which knowledge is imprecise or incomplete.

Convergence in analysis, either classical or fuzzy, implies that almost all elements of the sequence satisfy the convergence condition. As an example, in classical convergence, almost all elements of the sequence have to belong to an arbitrarily small neighborhood of the limit. The principal thought of statistical convergence is to take this condition easy and to demand validity of the convergence condition only for a majority of elements. As usual, statistics are concerned only about big quantities and majority is simply signify the concept “almost all” in pure mathematics. The reason is that statistics works with finite populations and samples, whereas pure mathematics is mostly concerned in infinite sets.

Statistical convergence was firstly examined by Fast [2]. In the wake of the study of ideal convergence defined by Kostyrko et al. [3], there has been comprehensive research to discover applications and summability studies of the classical theories. Ideal convergence became a noteworthy topic in summability theory after the studies of [4 –9].

Various types of convergence for sequences of functions, such as pointwise, equi-statistical (or, ideal) and uniform convergence was originated by Balcerzak et al. [10]. Ideal convergence of continuous functions was worked by Jasinski and Reclaw [11]. In the study [12], Egorov’s theorem for analytic P-ideals was proved.

Additionally, Duman and Orhan [13] examined convergence in μ-density and μ-statistical convergence of sequences of functions and presented μ-statistical pointwise convergence and μ-statistical uniform convergence.

Lacunary statistical convergence was firstly given by Fridy and Orhan [14]. The publication of the paper affected deeply all the scientific fields.

Matloka [15] identified convergence of a sequence of fuzzy numbers. Nanda [16] studied the sequences of fuzzy numbers and demonstrated that the set of all convergent sequences of fuzzy numbers create a complete metric space. Statistical convergence by utilizing fuzzy numbers was given by Nuray and Savaş [17]. By using fuzzy numbers, Aytar and Pehlivan [18] defined the statistical convergence of sequences. Pointwise statistical convergence sequences of fuzzy mappings studied by Altin et al. [19]. Some noteworthy results on this topic can be found in ([20 –34]). Kumar and Kumar [35] investigated $I$ -convergence of fuzzy numbers. Hazarika [36] examined lacunary $I$ -convergent sequence of fuzzy real numbers and gave some basic features.

Gong et al. [37] examined statistical convergence, equi-statistical convergence and uniformly statistical convergence of sequences of fuzzy valued functions. Hazarika ([38, 39]) gave ideal convergence versions of these concepts which were defined in [37] and established significant results. Kişi and Dündar [40] examined lacunary statistical convergence in measure for sequences of fuzzy valued functions.

The purpose of this study is to present some recent improvement in the theory of convergence in measure for sequences of fuzzy valued functions. We know that ideal convergence is more general than statistical convergence for sequences. This has focused us to study the lacunary convergence in measure for sequences of fuzzy valued functions further by using ideals and examined some features of this new type convergence. Our results were emphasized with examples.

Now, we recall some concepts and basic definitions used in the paper.

From the study of Zadeh [1], a fuzzy subset of T is a nonempty subset ${t, \bar{w} (t) : t \in T}$ of T × J (= [0, 1]) for some function $\bar{w} : T \to J$ . $\bar{w} : ℝ \to J$ is named a fuzzy number if it holds the following features:

(i) $\bar{w}$ is convex, i.e., $\bar{w} (t) \geq \bar{w} (s) Λ \bar{w} (r) = min {\bar{w} (s), \bar{w} (r)}$ , where s < t < r .

(ii) $\bar{w}$ is normal, i.e., there is an $t_{0} \in ℝ$ such that $\bar{w} (t_{0}) = 1 .$

(iii) $\bar{w}$ is upper semi-continuous, i.e., for all ɛ > 0, ${\bar{w}}^{- 1} ([0, a + ɛ))$ , for all a ∈ [0, 1] is open in the usual topology of $ℝ$ .

(iv) ${[\bar{w}]}^{0} = cl ({t \in ℝ : \bar{w} (t) \geq 0})$ is compact. Here cl denotes the closure operator.

We indicate the set of all fuzzy numbers by $F (ℝ)$ . The set $ℝ$ of real numbers can be included in $F (ℝ)$ if we take $\bar{r} \in F (ℝ)$ by $\bar{r} (t) = {\begin{matrix} 1, & if t = r, \\ 0, & if t \neq r . \end{matrix}$ For 0 < α ≤ 1, α-cut of $\bar{w}$ is given by ${[\bar{w}]}_{α} = {t \in ℝ : \bar{w} (t) \geq α} = [{\bar{w}}_{α}^{-}, {\bar{w}}_{α}^{+}]$ . In [41], the Hausdorff distance among $\bar{w}$ and $\bar{q}$ denoted by $D : F (ℝ) \times F (ℝ) \to [0, \infty)$ is given as follows: $\begin{matrix} D (\bar{w}, \bar{q}) = sup_{α \in [0, 1]} d ({[\bar{w}]}_{α}, {[\bar{q}]}_{α}) \\ = sup_{α \in [0, 1]} max {| {\bar{w}}_{α}^{-} - {\bar{w}}_{α}^{+} |, | {\bar{q}}_{α}^{-} - {\bar{q}}_{α}^{+} |}, \end{matrix}$ Here, d indicates Hausdorff metric.

For $K \subset ℕ$ and $j \in ℕ$ , δ_j (K) is named jth partial density of K, if $δ_{j} (K) = \frac{| K \cap {1, 2, . . ., j} |}{j} .$ If $\begin{matrix} δ (K) = lim_{n \to \infty} \frac{1}{n} | {k \leq n : k \in K} |, \\ (i . e ., δ (K) = lim_{j \to \infty} δ_{j} (K)) \end{matrix}$ exists, it is named the natural density of K. $Ψ = {K \subset ℕ : δ (K) = 0}$ is denoted the zero density set.

Let $K \subset ℕ$ and $r \in ℕ$ . $δ_{θ}^{r} (K)$ is named the rth partial lacunary density of K, if $δ_{θ}^{r} (K) = \frac{| K \cap I_{r} |}{h_{r}},$ where I_r = (k_r-1, k_r].

The number δ_θ (K) is denoted the lacunary density (θ-density) of K if $\begin{matrix} δ_{θ} (K) = lim_{r \to \infty} \frac{1}{h_{r}} | {k \in I_{r} : k \in K} |, \\ (i . e ., δ_{θ} (K) = lim_{r \to \infty} δ_{θ}^{r} (K)) \end{matrix}$ exists. Also, $Λ = {K \subset ℕ : δ_{θ} (K) = 0}$ is called to be zero density set.

Notable results about lacunary statistical convergence in measure for sequences of fuzzy valued functions were given by Kişi and Dündar in the study [40].

Take S≠ ∅. Then $\emptyset \neq I \subseteq P (S)$ is called to be an ideal on S iff (i) $\emptyset \in I$ , (ii) for each $A, B \in I$ one has $A \cup B \in I$ , (iii) for each $A \in I$ and B ⊆ A one has $B \in I$ . An ideal $I \subseteq P (S)$ is called to be non-trivial if $I \neq \emptyset$ and $S \notin I$ . A non-empty family of sets $F \subseteq P (S)$ is named to be filter on S iff (i) $\emptyset \notin F$ , (ii) for each $A, B \in F$ one has $A \cap B \in F$ , (iii) for each $A \in F$ and each B ⊇ A one has $B \in F$ . For every ideal $I$ , there is a filter $F (I)$ corresponding to $I$ i.e. $F (I) = {K \subset S : K^{c} \in I}$ , where K^c = S - K. We call that $I \subseteq P (S)$ is (i) an admissible ideal on S iff it includes all singletons, i.e., if it includes {{ y } : y ∈ S } (ii) maximal, if there isn’t any non-trivial ideal $J \neq I$ including $I$ as a subset (iii) called to be atranslation invariant ideal if ${k + 1 : k \in S} \in I$ , for any $S \in I$ .

(x_n) is said to be $I$ -convergent to ξ if for each ɛ > 0 the set $A (ɛ) = {n \in ℕ : | x_{n} - ξ | \geq ɛ}$ belongs to $I$ .

If we take ${I = I}_{δ} = {A \subseteq N : δ (A) = 0}$ where δ (A) indicate the asymptotic density of the set A, then ideal convergence coincides with statistical convergence.

Presume that $I \subseteq N$ is an admissible. Also, suppose that the ideals are proper ( $\neq P (ℕ)$ ) and include all finite sets. The ideals which includes of all finite sets is showed by Fin. If $A \notin I$ , then $I ↿ A$ , we say the restriction of $I$ to the set A, i.e., $I ↿$ $A = {B \subset A : B \in I}$ . If for each sequence ${(A_{n})}_{n \in ℕ}$ of sets from $I$ there is an $A \in I$ such that A_n \ A is finite for all n, then $I$ is a P-ideal.

By determining subsets of naturals with their characteristic functions, we equip $P (ℕ)$ with the Cantor space topology. Particularly, $I$ is analytic if it is a continuous image of a G_δ subset of the Cantor space. A nontrivial analytic P-ideal is the ideal of sets which has statistical density zero, i.e. $I_{δ} = {A \subseteq ℕ : lim_{j \to \infty} sup δ_{j} (A) = 0},$ Here, $δ_{j} (A) = \frac{| A \cap j |}{j}$ denotes the jth partial density of A. Additional example of an analytic P-ideal is $Φ \times ℕ$ . It is an ideal on $ℕ \times ℕ$ , which includes subsets of $ℕ \times ℕ$ , whose all vertical sections are finite.

A map $φ : P (ℕ) \to [0, \infty)$ is named submeasure on $ℕ$ if

(i) φ (Φ) = 0

(ii) φ (A) ≤ φ (A ∪ B) ≤ φ (A) + φ (B) for every $A, B \subset ℕ$

(iii) It is lower semicontinuous if for every $A \subseteq ℕ$ we get $φ (A) = lim_{n \to \infty} φ (A \cap n)$ .

Take submeasure $| | . | |_{φ} : P (ℕ) \to [0, \infty)$ for any lower semicontinuous submeasure on $ℕ$ . It is given by $| | A | |_{φ} = lim_{n \to \infty} sup φ (A \ n) = lim_{n \to \infty} φ (A \ n),$ where the second equality follows by the monotonocityof φ. Take $Exh (φ) = {A \subseteq ℕ : | | A | |_{φ} = 0} .$ It is obvious that Exh (φ) is an ideal for an optional submeasure φ .

Lemma 1.1. ([42]) Next situations are equivalent for an ideal $I \subset N$ . (a) $I$ is an analytic P-ideal, (b) $I = Exh (φ)$ for some lower semicontinuous submeasure φ on $ℕ$ .

Lemma 1.2. ([12]) Presume that $I = Exh (φ)$ for some lower semicontinuous submeasure. Then, $I$ is dense iff $lim_{n \to \infty} φ ({n}) = 0$ .

Lemma 1.3. ([12]) If $I$ is isomorphic to Φ×Fin or Fin then $lim_{m \in A} φ {m} = 0$ for some $A \notin I$ .

2 Main results

In the section, we presume that $\bar{h} : [a, b] \to F (ℝ)$ and ${\bar{h}}_{m} : [a, b] \to F (ℝ)$ are the fuzzy-valued function and a sequence of fuzzy-valued functions for all $m \in ℕ$ . We indicate SFVF and FVF in place of sequence of fuzzy-valued functions and fuzzy-valued function. We examine pointwise and uniformly $I_{θ}$ -convergence and equi-ideally lacunary convergence for a SFVF defined on [a, b].

Definition 2.1. A SFVF $({\bar{h}}_{m})$ is called to be pointwise $I_{θ}$ -convergent to a FVF $\bar{h}$ on [a, b], if ${\bar{h}}_{m} (y) \overset{p I_{θ}}{\to} \bar{h} (y)$ for each y ∈ [a, b], i.e. for all ɛ > 0, σ > 0, for all y ∈ [a, b] ${k \in ℕ : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ} \ k) \geq σ} \in I .$ This can be showed as ∀y ∈ [a, b], ∀ɛ > 0, ∀σ > 0, $\exists M_{y} \in I$ such that $φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ} \ k) < σ, \forall k \in ℕ \ M_{y} .$ We write ${\bar{h}}_{m} \overset{p I_{θ}}{\to} \bar{h}$ on [a, b].

Definition 2.2. A SFVF $({\bar{h}}_{m})$ is said to be uniformly $I_{θ}$ -convergent to a FVF $\bar{h}$ on [a, b], if for every ɛ > 0, σ > 0, ∀y ∈ [a, b] ${k \in ℕ : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ} \ k) \geq σ} \in I .$ This can be indicated as ∀y ∈ [a, b], ∀ɛ > 0, ∀σ > 0, $\exists M \in I$ such that $φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ} \ k) < σ, \forall k \in ℕ \ M .$ We write ${\bar{h}}_{m} \overset{u I_{θ}}{⇉} \bar{h}$ on [a, b].

Remark 2.1. If ${\bar{h}}_{m} \overset{u I_{θ}}{⇉} \bar{h}$ , then ${\bar{h}}_{m} \overset{p I_{θ}}{\to} \bar{h}$ . The converse is not necessarily true.

Example 2.1. For any y ∈ [0, 1], we think ${\bar{h}}_{m} (y) = {\begin{matrix} \bar{(\frac{my}{1 + m^{2} y^{2}})}, & for y \in [0, 1]; \\ \bar{0}, & otherwise . \end{matrix}$ Then, $({\bar{h}}_{m})$ pointwise $I_{θ}$ -converges to $\bar{h} = \bar{0}$ . But for all m ∈ I_r $\begin{matrix} sup_{y \in [0, 1]} D ({\bar{h}}_{m} (y), \bar{0}) & = sup_{y \in [0, 1]} \frac{my}{1 + m^{2} y^{2}} \\ = sup_{y \in [0, 1]} \frac{1}{\frac{1}{my} + my} = \frac{1}{2} \neq 0 . \end{matrix}$ Hence, $({\bar{h}}_{m})$ isn’t uniformly convergent and isn’t uniformly $I_{θ}$ -convergent to $\bar{0}$ on [0, 1].

Remark 2.2. ${\bar{h}}_{m} \overset{u I_{θ}}{⇉} \bar{h}$ iff $sup_{y \in [a, b]}$ $D ({\bar{h}}_{m} (y), \bar{h} (y)) \overset{p I_{θ}}{\to} 0$ .

Definition 2.3. A SFVF $({\bar{h}}_{m})$ is called equi-ideally lacunary convergent to a FVF $\bar{h}$ if ∀ɛ > 0, ∀σ > 0, such that ${k \in ℕ : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ} \ k) \geq σ}$ with regards to y ∈ [a, b] is uniformly convergent to zero function. It is indicated by ${\bar{h}}_{m} \overset{e I_{θ}}{↠} \bar{h}$ .

Example 2.2. For any y ∈ [0, 1] we think ${\bar{h}}_{m} (y) = {\begin{matrix} \bar{2^{m + 1} (y - \frac{1}{2^{m}})}, & for y \in [\frac{1}{2^{m}}, \frac{1}{2^{m - 1}} - \frac{1}{2^{m + 1}}]; \\ \bar{- 2^{m + 1} (y - \frac{1}{2^{m - 1}})}, & for y \in [\frac{1}{2^{m - 1}} - \frac{1}{2^{m + 1}}, \frac{1}{2^{m - 1}}]; \\ \bar{0}, & otherwise . \end{matrix}$ Then, $({\bar{h}}_{m})$ equi-ideally lacunary converges to a function $\bar{h} = \bar{0}$ . But $({\bar{h}}_{m})$ isn’t uniformly $I_{θ}$ -convergent to $\bar{0}$ on [0, 1].

Remark 2.3. ${\bar{h}}_{m} \overset{e I_{θ}}{↠} \bar{h}$ iff ∀ɛ > 0, ∀σ > 0, $\exists k \in ℕ$ , ∀y ∈ [a, b] such that $φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ} \ k) < σ .$

Remark 2.4. ${\bar{h}}_{m} \overset{u I_{θ}}{⇉} \bar{h} \Rightarrow {\bar{h}}_{m} \overset{e I_{θ}}{↠} \bar{h} \Rightarrow {\bar{h}}_{m} \overset{p I_{θ}}{\to} \bar{h}$ .

Proposition 2.1. Equi-ideally lacunary convergence is clearly defined with regards to α for α ∈ [0, 1].

Proof. To demonstrate contradiction, presume that there are two lower semicontinuous submeasures φ₁ and φ₂ such that Exh (φ₁) = Exh (φ₂) and ∀ɛ₁ > 0, $\exists k \in ℕ$ , ∀y ∈ X, $φ_{1} ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ_{1}} \ k) < ɛ_{1};$ ∀ɛ₂ > 0, $\exists k \in ℕ$ , ∀y ∈ X, $φ_{2} ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ_{2}} \ k) < ɛ_{2} .$ For all y ∈ X and for α ∈ [0, 1] we indicate $\begin{matrix} U (y) = {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ_{2}} \\ = {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} sup_{α \in [0, 1]} max {| {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} |, \\ | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} |} \geq ɛ_{2}} \end{matrix}$ and $\begin{matrix} U_{1} (y) = {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} | {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} | \geq ɛ_{2}}, \\ U_{2} (y) = {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} | \geq ɛ_{2}} . \end{matrix}$ We identify k₁ and y₁ such that $\begin{matrix} φ_{1} (U_{1} (y_{1}) \ k_{1}) < \frac{1}{2}, φ_{1} (U_{2} (y_{1}) \ k_{1}) < \frac{1}{2} \\ and φ_{2} (U_{1} (y_{1}) \ k_{1}) \geq ɛ_{2}, φ_{2} (U_{2} (y_{1}) \ k_{1}) \geq ɛ_{2} . \end{matrix}$ Since submeasure φ₂ is lower continuous, we are able to find $k_{1}^{'}$ with $\begin{matrix} φ_{2} ((U_{1} (y_{1}) \ k_{1}) \cap k_{1}^{'}) \geq \frac{ɛ_{2}}{2}, \\ φ_{2} ((U_{2} (y_{1}) \ k_{1}) \cap k_{1}^{'}) \geq \frac{ɛ_{2}}{2} . \end{matrix}$ Presume that we have found $k_{i}^{'}$ . Take $k_{i + 1} > k_{i}^{'}$ and y_i+1 such that $\begin{matrix} φ_{1} (U_{1} (y_{i + 1}) \ k_{i + 1}) < \frac{1}{2^{i + 1}}, φ_{1} (U_{2} (y_{i + 1}) \ k_{i + 1}) < \frac{1}{2^{i + 1}} and \\ φ_{2} (U_{1} (y_{i + 1}) \ k_{i + 1}) \geq ɛ_{2}, φ_{2} (U_{2} (y_{i + 1}) \ k_{i + 1}) \geq ɛ_{2} . \end{matrix}$ Moreover by the lower continuity of φ₂ we find $k_{i + 1}^{'}$ with $\begin{matrix} φ_{2} ((U_{1} (y_{i + 1}) \ k_{i + 1}) \cap k_{i + 1}^{'}) \geq \frac{ɛ_{2}}{2}, \\ φ_{2} ((U_{2} (y_{i + 1}) \ k_{i + 1}) \cap k_{i + 1}^{'}) \geq \frac{ɛ_{2}}{2} . \end{matrix}$ We take $\begin{matrix} P = ⋃_{i \in ℕ} ((U_{1} (y_{i}) \ k_{i}) \cap k_{i}^{'}) and \\ R = ⋃_{i \in ℕ} ((U_{2} (y_{i}) \ k_{i}) \cap k_{i}^{'}) . \end{matrix}$ Then, we get $\begin{matrix} | | P | |_{φ_{1}} & = & lim_{i \to \infty} φ_{1} (P \ k_{i}) \\ \leq & lim_{i \to \infty} \sum_{n = i}^{\infty} φ_{1} (U_{1} (y_{i}) \cap {k_{i}, k_{i}^{'}}) \\ \leq & lim_{i \to \infty} \sum_{n = i}^{\infty} \frac{1}{2^{n}} = 0 \end{matrix}$ and $\begin{matrix} | | R | |_{φ_{1}} & = & lim_{i \to \infty} φ_{1} (R \ k_{i}) \\ \leq & lim_{i \to \infty} \sum_{n = i}^{\infty} φ_{1} (U_{2} (y_{i}) \cap {k_{i}, k_{i}^{'}}) \\ \leq & lim_{i \to \infty} \sum_{n = i}^{\infty} \frac{1}{2^{n}} = 0 . \end{matrix}$ So, we get P ∈ Exh (φ₁) and R ∈ Exh (φ₁). On the other side, we get $\begin{matrix} | | P | |_{φ_{2}} = lim_{i \to \infty} φ_{2} (P \ k_{i}) \geq \frac{ɛ_{2}}{2} > 0 \\ and | | R | |_{φ_{2}} = lim_{i \to \infty} φ_{2} (R \ k_{i}) \geq \frac{ɛ_{2}}{2} > 0 \end{matrix}$ which is a contradiction. This contradicts imply the proof the theorem.□

Proposition 2.2. A SFVF $({\bar{h}}_{m})$ is equi-lacunary convergent to FVF $\bar{h}$ iff it is equi-ideally lacunary convergent to $\bar{h}$ with regards to $I_{δ}$ .

Proof. Take $I_{δ} = Exh (φ)$ . Here $φ (A) = sup_{j} δ_{j} (A)$ for $A \subset ℕ$ . Hence, following results conclude the proof. (a) If δ_j (A) < ɛ for j ≥ k then $sup_{j} δ_{j} (A \ k) \leq ɛ$ (b) If $sup_{j} δ_{j} (A \ k) \leq ɛ$ then δ_j (A) ≤ δ_j (A \ k) + δ_j (k) ≤ 2ɛ for $j \geq \frac{k}{ɛ}$ .□

Proposition 2.3. ([39]) Let $({\bar{h}}_{m})$ be a SFVF and $\bar{h}$ be a FVF. Then ${\bar{h}}_{m} \overset{e Fin}{↠} \bar{h} \Rightarrow {\bar{h}}_{m} \overset{e Fin}{⇉} \bar{h}$ .

Proposition 2.4. Presume that $I = Exh (φ)$ for some lower semicontinuous submeasure. Then $I$ is isomorphic to Fin or φ×Fin iff for any SFVF equi-ideally lacunary convergence and uniform $I_{θ}$ -convergence are equivalent.

Proof. Take $I = φ \times$ Fin and presume that ${\bar{h}}_{m} \overset{e I_{θ}}{↠} \bar{h}$ . To indicate that ${\bar{h}}_{m} \overset{u I_{θ}}{⇉} \bar{h}$ . If possible assume that $⇁ ({\bar{h}}_{m} \overset{u I_{θ}}{⇉} \bar{h})$ . Then, for all y ∈ X and for some ɛ > 0, we get $A = {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq \frac{1}{2}} \notin I .$ There is B ⊂ A, $B \notin I$ such that $I ↾ B =$ Fin. Hence, we get ${\bar{h}}_{m} \overset{e I_{θ} ↾ B}{↠} \bar{h}$ and $⇁ ({\bar{h}}_{m} \overset{u I_{θ} ↾ B}{⇉} \bar{h})$ which contradicts the Proposition 2.3. Next we consider any analytic P-ideal $I = Φ \times$ Fin which isn’t isomorphic to Φ×Fin or Fin. From Lemma 1.3, we get $lim_{m \in A} φ ({m}) = 0$ for some $A \notin I$ . Take ${\bar{h}}_{m}$ as the characteristic function of ${\frac{1}{m}}$ for m ∈ A and zero function otherwise. For each y ∈ X, we get ${r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq \frac{1}{2}} = A \notin I$ so, $⇁ ({\bar{h}}_{m} \overset{u I_{θ}}{⇉} \bar{h})$ . We demonstrate that ${\bar{h}}_{m} \overset{e I_{θ}}{↠} \bar{h}$ . Take ɛ > 0 and $n \in ℕ$ such that φ ({ m }) < ɛ for m > n. For any α ∈ [0, 1], the set $\begin{matrix} B_{y} = {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} sup_{α \in [0, 1]} max {| {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} |, \\ | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} |} \geq ɛ} \end{matrix}$ includes at most one element for each y ∈ [0, 1]. Hence, μ (B_y \ n) < ɛ and ${\bar{h}}_{m} \overset{e I_{θ}}{↠} \bar{h}$ .□

Proposition 2.5. If $I$ is a translation invariant ideal and $I_{θ}$ - $lim_{k \to \infty} {\bar{h}}_{k} (y) = \bar{h} (y)$ , then $I_{θ}$ - $lim_{k \to \infty} {\bar{h}}_{k + 1} (y) = \bar{h} (y)$ for each y ∈ [a, b].

Theorem 2.1. Let $I$ be an admissible ideal. Let $({\bar{h}}_{m})$ be a SFVF and $\bar{h}$ be a FVF determined on [a, b]. For each y ∈ [a, b], if ${\bar{h}}_{m} (y) \overset{p I_{θ}}{\to} \bar{h} (y)$ , then ${[{\bar{h}}_{m} (y)]}_{α} \overset{p I_{θ}}{\to} {[\bar{h} (y)]}_{α}$ with regards to α.

Proof. Presume that for each y ∈ [a, b], ${\bar{h}}_{m} (y) \overset{p I_{θ}}{\to} \bar{h} (y)$ . Take ɛ > 0. For each y ∈ X, then there exists $n = n (y, ɛ) \in ℕ$ such that ∀ɛ > 0, ∀σ > 0,

$φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ} \ k) < σ$ (2.1) for all k ≥ n. This gives that for any α ∈ [0, 1], ∀ɛ > 0, ∀σ > 0, $\begin{matrix} {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} sup_{α \in [0, 1]} max {| {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} |, \\ | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} |} \geq ɛ} \ k) < σ \end{matrix}$ (2.2) for all k ≥ n. For any α ∈ [0, 1], we get ∀ɛ > 0, ∀σ > 0, $φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} | {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} | \geq ɛ} \ k) < σ$ (2.3) for all k ≥ n. Also ∀ɛ > 0, ∀σ > 0, $φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} | \geq ɛ} \ k) < σ$ (2.4) for all k ≥ n. From relation (2.1) we have ∀ɛ > 0, ∀σ > 0, $\begin{matrix} {k \in ℕ : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq ɛ} \ k) \geq σ} \\ \subset ℕ \cup {1, 2, . . ., n - 1} . \end{matrix}$ Again from (2.2) we have ∀ɛ > 0, ∀σ > 0, the following set $\begin{matrix} {k \in ℕ : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} sup_{α \in [0, 1]} max {| {({\bar{h}}_{m} (y))}_{α}^{-} - {\bar{h}}_{α}^{-} (y) |, \\ | {({\bar{h}}_{m} (y))}_{α}^{+} - {\bar{h}}_{α}^{+} (y) |} \geq ɛ} \ k) \geq σ} \end{matrix}$ is subset of $ℕ \cup {1, 2, . . ., n - 1}$ . For any α ∈ [0, 1]. Then from (2.3) and (2.4) we get ∀ɛ > 0, ∀σ > 0, $\begin{matrix} {k \in ℕ : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} | {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} | \geq ɛ} \ k) \geq σ} \\ \subset ℕ \cup {1, 2, . . ., n - 1} \end{matrix}$ and $\begin{matrix} {k \in ℕ : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} | \geq ɛ} \ k) \geq σ} \\ \subset ℕ \cup {1, 2, . . ., n - 1} . \end{matrix}$ Consider $I$ as admissible ideal, hence we obtain $ℕ \cup {1, 2, . . ., n - 1} \in I$ and as a result we have for each ɛ > 0 and for every σ > 0, $\begin{matrix} {k \in ℕ : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} | {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} | \geq ɛ} \ k) \geq σ} \\ \in I \end{matrix}$ and $\begin{matrix} {k \in ℕ : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} | \geq ɛ} \ k) \geq σ} \\ \in I . \end{matrix}$ Hence, for each y ∈ [a, b] we obtain ${[{\bar{h}}_{m} (y)]}_{α} \overset{p I_{θ}}{\to} {[\bar{h} (y)]}_{α}$ with regards to α.□

The next result is the lacunary ideal form of Egorov’s theorem for the SFVF.

Presume that X is a finite measurable set. We indicate $L^{0}$ , the set of fuzzy valued measurable functions defined almost everywhere on X.

Take A as any finite subset of $ℕ$ , consider $[A, n] = {B \subset ℕ : B \cap n = A}$ . The sets [A, n] create a base of the Cantor-set topology on $P (ℕ)$ .

Theorem 2.2. Let $I$ be an analytic P-ideal and $(X, M, μ)$ be a finite measure space. Let the FVF $\bar{h}$ and the SFVF $({\bar{h}}_{m})$ be measurable and defined on almost everywhere on X. Presume also that ${\bar{h}}_{m} \overset{p I_{θ}}{\to} \bar{h}$ almost everywhere on X. Then, for each ɛ > 0 there is a subset $A$ of X such that $μ (X \ A) < ɛ$ and ${\bar{h}}_{m ∣ A} \overset{e I_{θ}}{↠} {\bar{h}}_{∣ A}$ on $A$ .

Proof. Let $(X, M, μ)$ be a finite measure space. We presume that for all $m \in ℕ,$ $\bar{h}, {\bar{h}}_{m}$ are defined everywhere on X and also assume that ${\bar{h}}_{m} (y) \overset{p I_{θ}}{\to} \bar{h} (y)$ for all y ∈ X. For any fix $k, σ \in ℕ$ , we consider the set $\begin{matrix} E_{k, σ} = {y \in X : φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq \frac{1}{σ}} \ k) \\ < \frac{1}{σ}} . \end{matrix}$ We indicate that the set E_k,σ is measurable. For this, we demonstrate that the complement of every E_k,σ is measurable. $\begin{matrix} \begin{matrix} X \ E_{k, σ} \\ = {y \in X : ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq \frac{1}{σ}} \ k) \\ \in φ^{- 1} ((\frac{1}{σ}, \infty))} . \end{matrix} \end{matrix}$ Since φ is lower semicontinuous, there exist sets [A_i, n_i] such that $\begin{matrix} \begin{matrix} X \ E_{k, σ} \\ = {y \in X : ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq \frac{1}{σ}} \ k) \\ \in ⋃_{i \in ℕ} [A_{i}, n_{i}]} \\ = ⋃_{i \in ℕ} \overset{n_{i}}{⋂_{j = k}} {y \in X : (r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq \frac{1}{σ})}^{χ_{A_{i} (j)}} . \end{matrix} \end{matrix}$ Since $({\bar{h}}_{m})$ and $\bar{h}$ are measurable, the right hand side set in the last relation is measurable and hence the set X \ E_k,σ is measurable. For each $σ \in ℕ$ we get E_k,σ ⊂ E_k+1,σ and $X = ⋃_{k = 1}^{\infty} E_{k, σ}$ . As a result $μ (X) = lim_{r \to \infty} μ (E_{k, σ})$ . Take ɛ > 0. For each $σ \in ℕ$ , let $k (σ) \in ℕ$ be such that $μ (X \ E_{k (σ), σ}) < \frac{ɛ}{2^{σ}}$ . We take $A_{0} = ⋃_{σ = 1}^{\infty} (X \ E_{k (σ), σ})$ . Then, we have $μ (A_{0}) \leq \sum_{σ = 1}^{\infty} μ (X \ E_{k (σ), σ}) < ɛ$ . Let $A = X \ A_{0} = ⋂_{σ = 1}^{\infty} E_{k (σ), σ}$ . Then, we obtain μ (X \ A) = μ (A₀) < ɛ. Hence, we get ∀σ > 0, $\exists k (σ) \in ℕ$ , ∀y ∈ A, $φ ({r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq \frac{1}{σ}} \ k) < \frac{1}{σ} .$ This gives that ${\bar{h}}_{m ∣ A} \overset{e I_{θ}}{↠} {\bar{h}}_{∣ A}$ on $A$ .□

Corollary 2.1. Let $(X, M, μ)$ be a finite measure space. Suppose that the FVF $\bar{h}$ and SFVF $({\bar{h}}_{m})$ are measurable and defined almost everywhere on X. Then, ${\bar{h}}_{m} \overset{p I_{θ}}{\to} \bar{h}$ almost everywhere on X iff there exists a sequence (A_m) of sets on $M$ such that ${\bar{h}}_{m ∣ A_{m}} \overset{e I_{θ}}{↠} {\bar{h}}_{∣ A_{m}}$ on A_m for all m and $μ (X \ ⋃_{m \in ℕ} A_{m}) = 0$ .

Proof. To show “⇒”, think $ɛ = \frac{1}{m} (m \in ℕ)$ in Theorem 2.2. The part " ⇐ " follows from ${\bar{h}}_{m ∣ A_{m}} \overset{e I_{θ}}{↠} {\bar{h}}_{∣ A_{m}} \Rightarrow {\bar{h}}_{m ∣ A_{m}} \overset{p I_{θ}}{\to} {\bar{h}}_{∣ A_{m}}$ for all m.□

Now, we determine the $I_{θ}$ -convergence in measure for a SFVF and obtain some results.

Definition 2.4. Let $({\bar{h}}_{m}) \in L^{0}$ , $\bar{h} \in L^{0}$ and X be a finite measurable space. A SFVF $({\bar{h}}_{m})$ is called to be $I_{θ}$ -convergent in measure to a FVF $\bar{h}$ , in symbol, ${\bar{h}}_{m} \overset{μ I_{θ}}{\to} \bar{h}$ , if $μ ({y \in X : D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq η})$ is $I_{θ}$ -convergent to zero for every η > 0 and all m ∈ I_r. We give this notion as the following formula: ∀q > 0, ∀ η > 0, ${r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ ({y \in X : D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq η}) > q} \in I .$ Here, we can write η = q or $η = \frac{1}{s}$ , $s \in ℕ .$

Proposition 2.6. Let $(X, M, μ)$ be a finite measure space. Presume that $({\bar{h}}_{m}) \in L^{0}$ , $\bar{h} \in L^{0}$ . Then ${\bar{h}}_{m} \overset{u I_{θ}}{⇉} \bar{h} \Rightarrow {\bar{h}}_{m} \overset{μ I_{θ}}{\to} \bar{h} .$

Proof. We assume that ${\bar{h}}_{m} \overset{u I_{θ}}{⇉} \bar{h}$ . Take η > 0. Then, there is a set $T \in I$ such that $D ({\bar{h}}_{m} (y), \bar{h} (y)) < η$ for all m ∉ T, y ∈ X. Thus, we get $\begin{matrix} {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ ({y \in X : D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq η}) > η} \\ \subset {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ ({y \in X : D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq η}) \neq \emptyset} \\ \subset T \in I . \end{matrix}$ This indicates that ${\bar{h}}_{m} (y) \overset{μ I_{θ}}{\to} \bar{h} (y)$ .□

Theorem 2.3. Let $({\bar{h}}_{m}) \in L^{0}$ , $\bar{h} \in L^{0}$ and X be a finite measurable space. A SFVF $({\bar{h}}_{m})$ is $I_{θ}$ -convergent in measure to FVF $\bar{h}$ iff ${[{\bar{h}}_{m} (y)]}_{α}$ is $I_{θ}$ -convergent in measure to ${[\bar{h} (y)]}_{α}$ with regards to α.

Proof. Presume that $({\bar{h}}_{m})$ is $I_{θ}$ -convergent in measure to $\bar{h}$ . Then, we get $μ ({y \in X : D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq η})$ is $I_{θ}$ -convergent to zero for every η > 0 and for each m ∈ I_r. i.e., ${r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ ({y \in X : D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq η}) \geq η} \in I .$ Thus, we get $\begin{matrix} {\begin{matrix} r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} \\ μ ({y \in X : sup_{α \in [0, 1]} max {| {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} |, \\ | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} |} \geq η}) \geq η \end{matrix}} \in I . \end{matrix}$ Therefore, for α ∈ [0, 1], one obtains $\begin{matrix} {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ ({y \in X : sup_{α \in [0, 1]} | {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} | \\ \geq η}) \geq η} \in I \end{matrix}$ and $\begin{matrix} {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ ({y \in X : sup_{α \in [0, 1]} | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} | \\ \geq η}) \geq η} \in I, \end{matrix}$ which yields that $μ ({y \in X : sup_{α \in [0, 1]} | {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} | \geq η}) \overset{μ I_{θ}}{\to} 0$ and $μ ({y \in X : sup_{α \in [0, 1]} | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} | \geq η}) \overset{μ I_{θ}}{\to} 0 .$ Hence, ${[{\bar{h}}_{m} (y)]}_{α}$ is $I_{θ}$ -convergent in measure to ${[\bar{h} (y)]}_{α}$ with regards to α. Next, we assume that ${[{\bar{h}}_{m} (y)]}_{α}$ is $I_{θ}$ -convergent in measure to ${[\bar{h} (y)]}_{α}$ with regards to α. For each η > 0, we get $μ ({y \in X : sup_{α \in [0, 1]} | {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} | \geq η}) \overset{μ I_{θ}}{\to} 0$ and $μ ({y \in X : sup_{α \in [0, 1]} | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} | \geq η}) \overset{μ I_{θ}}{\to} 0 .$ Thus, we obtain $\begin{matrix} {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ ({y \in X : sup_{α \in [0, 1]} | {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} | \geq η}) \\ \geq η} \in I \end{matrix}$ and $\begin{matrix} {r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ ({y \in X : sup_{α \in [0, 1]} | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} | \geq η}) \\ \geq η} \in I . \end{matrix}$ From the last two relations, we get ${\begin{matrix} r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} \\ μ ({y \in X : sup_{α \in [0, 1]} max {| {({\bar{h}}_{m} (y))}_{α}^{-} - {(\bar{h} (y))}_{α}^{-} |, \\ | {({\bar{h}}_{m} (y))}_{α}^{+} - {(\bar{h} (y))}_{α}^{+} |} \geq η}) \geq η \end{matrix}} \in I,$ which gives that ${r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ ({y \in X : D ({\bar{h}}_{m} (y), \bar{h} (y)) \geq η}) \geq η} \in I .$ This means that $({\bar{h}}_{m})$ is $I_{θ}$ -convergent in measure to FVF $\bar{h}$ .□

Definition 2.5. Let $({\bar{h}}_{m})$ be a sequence of fuzzy valued measurable functions in $L^{0}$ . Then $({\bar{h}}_{m})$ is called to be $I_{θ}$ -Cauchy in measure if there is an integer $n \in ℕ$ such that ∀ɛ > 0, ∀ η > 0, ${r \in ℕ : \frac{1}{h_{r}} \sum_{m \in I_{r}} μ (D ({\bar{h}}_{m}, {\bar{h}}_{n}) \geq ɛ) \geq η} \in I .$

Remark 2.5. Let $({\bar{h}}_{m})$ be a sequence of fuzzy valued measurable functions in $L^{0}$ . Then, we get

(a) $E = {y \in [a, b] : I_{θ} - lim_{m \to \infty} {\bar{h}}_{m} (y) exists}$ is measurable.

(b) $\bar{h} (y) = {\begin{matrix} I_{θ} - lim_{m \to \infty} {\bar{h}}_{m} (y) & if the limit exists, \\ \bar{0} & otherwise . \end{matrix}$ is measurable function.

Theorem 2.4. Let $({\bar{h}}_{m})$ be a sequence of fuzzy valued measurable functions in $L^{0}$ . Following situations are equivalent:

(a) $({\bar{h}}_{m})$ is $I_{θ}$ -Cauchy in measure.

(b) there is an $\bar{h} \in L^{0}$ such that ${\bar{h}}_{m} \overset{μ I_{θ}}{\to} \bar{h}$ .

Proof. (a) ⇒ (b) Presume that $({\bar{h}}_{m})$ is $I_{θ}$ -Cauchy in measure. We indicate that there is a subsequence ${({\bar{g}}_{i})}_{i \in I_{r}}$ of $({\bar{h}}_{m})$ such that ${r \in ℕ : \frac{1}{h_{r}} \sum_{i \in I_{r}} μ (D ({\bar{g}}_{i}, {\bar{g}}_{i + 1}) \geq \frac{1}{2^{i}}) \leq \frac{1}{2^{i}}} \in F .$

Define $\begin{matrix} E_{i} = {r \in ℕ : \frac{1}{h_{r}} \sum_{i \in I_{r}} μ (D ({\bar{g}}_{i}, {\bar{g}}_{i + 1})) \geq \frac{1}{2^{i}}} and \\ H_{k} = ⋃_{i = k}^{\infty} E_{i} . \end{matrix}$ Then, we obtain $μ (E_{i}) \leq \frac{1}{2^{i}} and μ (H_{k}) \leq \sum_{i = k}^{\infty} \frac{1}{2^{i}} = \frac{1}{2^{k - 1}} .$ If we put $P = ⋂_{k = 1}^{\infty} H_{k} = ⋂_{k = 1}^{\infty} ⋃_{i = k}^{\infty} E_{i} = lim_{i \to \infty} sup E_{i},$ then we get P ⊆ H_k for every k and μ (P) = 0. Now, fix any y ∉ P. Then y ∉ H_k for some k, and so y ∉ E_i for every i ≥ k. So, for any l ≥ i ≥ k we obtain $D ({\bar{g}}_{i} (y), {\bar{g}}_{l} (y)) \leq \sum_{j = i}^{l - 1} D ({\bar{g}}_{j + 1} (y), {\bar{g}}_{j} (y)) \leq \sum_{j = i}^{l - 1} \frac{1}{2^{j}} \leq \frac{1}{2^{i - 1}} .$

As a result, the SFVF (g_i (y)) is Cauchy. From the completeness of $F (ℝ)$ , hence it is convergent to a FVF. If we take $\bar{h} (y) = {\begin{matrix} I_{θ} - lim_{i \to \infty} {\bar{g}}_{i} (y) & if the limit exists, \\ \bar{0} & otherwise . \end{matrix}$ Then $\bar{h}$ is measurable and from the existence of limit for each y ∉ P we get that ${\bar{h}}_{m} \overset{p I_{θ}}{\to} \bar{h}$ almost everywhere. Now we demonstrate that (g_i) $I_{θ}$ -converges in measure to $\bar{h}$ . Fix any i. If y ∉ H_i, then we have $D ({\bar{g}}_{i} (y), {\bar{g}}_{l} (y)) \leq \frac{1}{2^{i - 1}}$ for all l ≥ i. Hence $\begin{matrix} D ({\bar{g}}_{i} (y), \bar{h} (y)) \\ \leq D ({\bar{g}}_{i} (y), I_{θ} - lim_{l \to \infty} {\bar{g}}_{l} (y)) \\ = I_{θ} - lim_{l \to \infty} (D ({\bar{g}}_{i} (y), {\bar{g}}_{l} (y))) \leq \frac{1}{2^{i - 1}} . \end{matrix}$ Therefore, ${r \in ℕ : \frac{1}{h_{r}} \sum_{i \in I_{r}} D ({\bar{g}}_{i}, \bar{h}) \geq \frac{1}{2^{i - 1}}} \subseteq H_{k},$ so we obtain $μ (D ({\bar{g}}_{i}, \bar{h}) \geq \frac{1}{2^{i - 1}}) \leq μ (H_{k}) \leq \frac{1}{2^{i - 1}} \to 0$ as i→ ∞ It implies that ${\bar{g}}_{i} \overset{μ I_{θ}}{\to} \bar{h}$ . So, we have indicated that $({\bar{g}}_{l})$ is a subsequence of $({\bar{h}}_{m})$ that $I_{θ}$ -converges in measure to $\bar{h}$ . Unite this with the fact that $({\bar{h}}_{m})$ is ideally lacunary Cauchy in measure to demonstrate that ${\bar{h}}_{m} \overset{μ I_{θ}}{\to} \bar{h} .$ (b) ⇒ (a) It is obvious, so omitted.□

References

Zadeh

L.A.

, Fuzzy sets, Inform Control 8 (1965), 338–353.

Fast

, Sur la convergence statistique, Colloq Math 2 (1951), 241–244.

Kostyrko

, Šalát

and Wilczyski

I

-convergence, Real Anal Exchange 26(2) (2000), 669–686.

Cartan

, Filters et ultrafilters, C R Acad Sci Paris 205 (1937), 777–779.

Kostyrko

, Macaj

, Šalát

and Sleziak

I

-convergence and extremal

I

-limit points, Math Slovaca 55 (2005), 443–464.

Šalát

, Tripathy

B.C.

and Ziman

, On some properties of

I

-convergence, Tatra Mt Math Publ 28 (2004), 279–286.

Hazarika

, Mohiuddine

S.A.

, Ideal convergence of random variables, J Funct Spaces 2013, 7. Article ID 673148249.

Tripathy

B.C.

, Hazarika

, Choudhary

, Lacunary

I

-Convergent Sequences, Kyungpook Math J 52 (2012), 473–482.

Das

, Savaş

and Ghosal

S.Kr.

, On generalizations of certain summability methods using ideals, Appl Math Lett 24 (2011), 1509–1614.

10.

Balcerzak

, Dems

, Komisarski

, Statistical convergence and ideal convergence for sequences of functions, J Math Anal Appl 328 (2007), 715–729.

11.

Jasinski

, Reclaw

, Ideal convergence of continuous functions, Topology Appl 153 (2006), 3511–3518.

12.

Mrozek

, Ideal version of Egorov’s theorem for analytic P-ideals, J Math Anal Appl 390 (2009), 452–458.

13.

Duman

, Orhan

, μ-Statistically convergent function sequences, Czechoslovak Math J, 54(129) (2004), 413–422.

14.

Fridy

J.A.

, Orhan

, Lacunary statistical convergence, Pacific J Math 160(1) (1993), 43–51.

15.

Matloka

, Sequences of fuzzy numbers, Busefal 28 (1986), 28–37.

16.

Nanda

, On sequences of fuzzy numbers, Fuzzy Sets and Systems 33 (1989), 123–126.

17.

Nuray

, Savaş

, Statistical convergence of sequences of fuzzy numbers, Math Slovaca 45(3) (1995), 269–273.

18.

Aytar

, Pehlivan

, Statistical convergence of sequences of fuzzy numbers and sequences of α-cuts, Int J Gen Syst 37(2) (2008), 231–237.

19.

Altin

, Et

, Tripathy

B.C.

, On pointwise statistical convergence sequences of fuzzy mappings, J Fuzzy Math 15(2) (2007), 425–433.

20.

Altinok

, Altin

, Işık

, Statistical convergence and strong p-Cesàro summability of order β in sequences of fuzzy numbers, Iran J Fuzzy Syst 9(2) (2012), 65–75.

21.

Altinok

, Et

, Statistical convergence of order (β, γ) for sequences of fuzzy numbers, Soft Comput 23 (2019), 6017–6022.

22.

Çınar

, Karakaş

and Et

, On pointwise and uniform statistical convergence of order α for sequences of functions, Fixed Point Theory Appl {33 (2013), 11 pages.

23.

Mohiuddine

S.A.

, Asiri

, Hazarika

, Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems, Int J Gen Syst 48(5) (2019), 492–506.

24.

Mohiuddine

S.A.

, Hazarika

, Alotaibi

, On statistical convergence of double sequences of fuzzy valued functions, J Intell Fuzzy Syst 32 (2017), 4331–4342.

25.

Nuray

, Lacunary statistical convergence of sequences of fuzzy numbers, Fuzzy Sets and Systems 99 (1998), 353–355.

26.

Savaş

, On statistically convergent sequences of fuzzy numbers, Inform Sci 137(1-4) (2001), 277–282.

27.

Şençimen

and Pehlivan

, Statistical convergence in fuzzy normed linear spaces, Fuzzy Sets and Systems 159 (2008), 361–370.

28.

Turkmen

M.R.

, Dündar

, On lacunary statistical convergence of double sequences and some properties in fuzzy normed spaces, J Intell Fuzzy Systems 36(2) (2019), 1683–1690.

29.

Turkmen

M.R.

, Cinar

, λ-statistical convergence in fuzzy normed linear spaces, J Intell Fuzzy Systems 34(6) (2018), 4023–4030.

30.

Turkmen

M.R.

, Cinar

, Lacunary statistical convergence in fuzzy normed linear spaces, Appl Comput Math 6(5) (2017), 233–237.

31.

Dündar

, Turkmen

M.R.

, On

I_{2}

-convergence and

I_{2}^{*}

-convergence of double sequences in fuzzy normed spaces, Konuralp J Math 7(2) (2019), 405–409.

32.

Turkmen

M.R.

, On lacunary statistical convergence and some properties in fuzzy-normed spaces, i-Manager’s J Math 7(3) (2018), 1–9.

33.

Dündar

, Turkmen

M.R.

, Pancaroğlu Akın

, Regularly ideal convergence of double sequences in fuzzy normed spaces, Bull Math Anal Appl 12(2) (2020), 12–26.

34.

Turkmen

M.R.

, On

I_{θ_{2}}

-convergence in fuzzy normed spaces, J Inequal Appl 2020 (2020), 127. 10.1186/s13660-020-02387-9.

35.

Kumar

, Kumar

, On the ideal convergence of sequences of fuzzy numbers, Inform Sci 178 (2008), 4670–4678.

36.

Hazarika

, Fuzzy real valued lacunary

I

-convergent sequences, Appl Math Lett 25 (2012), 466–470.

37.

Gong

, Zhang

, Zhu

, The statistical convergence for sequences of fuzzy-number-valued functions, Inform Sci 295 (2015), 182–195.

38.

Hazarika

, Pointwise ideal convergence and uniformly ideal convergence of sequences of fuzzy valued functions, J Intell Fuzzy Systems 32(3) (2017), 2665–2677.

39.

Hazarika

, On ideal convergence in measure for sequences of fuzzy valued functions, J Intell Fuzzy Systems 35(5) (2018), 5729–5740.

40.

Kişi

Ö.

and Dündar

, Lacunary statistical convergence in measure for sequences of fuzzy valued functions, Konuralp J Math 8(2) (2020), 252–262.

41.

Negoita

C.V.

, Ralescu

, Applications of fuzzy sets to systems analysis, (1975), Wiley, New York.

42.

Solecki

, Analytic ideals and their applications, Ann Pure Appl Logic 99(1-3) (1999), 51–72.