On ideal convergence in measure for sequences of fuzzy valued functions

Abstract

We introduce the notions of pointwise and uniform ideal convergence and kind of convergence lying between aforementioned convergence methods, namely, equi-ideal convergence of sequences of fuzzy valued functions and obtain various results related to these kinds of convergence and their representations of sequences of α-level cuts. We prove the ideal version of the Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space $(X, M, μ) .$ We also define the notion of ideal convergence in measure for sequences of fuzzy valued functions and prove some interesting results.

Keywords

Fuzzy valued function ideal convergence Egorov’s theorem convergence in measure

1 Introduction

In analysis, either in classical or fuzzy, the convergence sequences means that almost all elements of the sequence have to belong to an arbitrary small neighborhood of the limit. The main purpose of the statistical convergence is to relax this condition and to demand validity of the convergence criterion only for a majority of the elements. As always, statistics are concerned only about the big quantities and the term “majority” is simply imply the concept “almost all” in the classical analysis. As the set of real numbers can be embedded in the set of all fuzzy numbers and statistical convergence of reals can be considered as a special case of fuzzy numbers. Since the set of all fuzzy numbers is partially ordered and does not carry the group structure, so most of the results in real sequences may not be valid in fuzzy setting. Therefore the theory is not a trivial extension of what has been known in real cases.

The notion of statistical convergence was introduced by Fast [8] and Steinhaus [28] and reintroduced by Schoenberg [26] and studied some basic properties of statistical convergence, and statistical convergence as a summability method. Fridy [9] and Šalát [23] gave characterizations of statistical convergence of real number sequences. The notion of the ideal convergence is the dual (equivalent) to the notion of filter convergence introduced by Cartan [7] in the year 1937. The notion of the filter convergence is a generalization of the classical notion of convergence of a sequence and it has been an important tool in general topology and functional analysis. Kostyrko et al. [14] and Nuray and Ruckle [19] independently studied in details about the notion of ideal convergence which is based on the structure of the admissible ideal I of subsets of natural numbers $ℕ .$ Later on it was further investigated by many authors, e.g. Šalát et al. [24], Hazarika and Mohiuddine [11], and references therein. Balcerzak et al. [4] discussed the statistical convergence and ideal convergence for sequences of real valued functions. Jasinski and Recław [13] introduced the concepts of ideal convergence of continuous functions. In [17] Mrożek proved the Egorov’s theorem for analytic P-ideals. Considering the uncertainty of data and information in a specific modeling process, this uncertainty was usually represented by a fuzzy number [18]. Şavas [25] proved the characterization theorem for the sequence of fuzzy numbers. Kumar and Kumar [15] introduced the concept of ideal convergence of fuzzy numbers and proved some classical theorems in fuzzy settings. Aytar and Pehlvian [3] discussed the statistical convergence of sequences of fuzzy numbers and sequences of α-cuts. Altin et al. [1] introduced the concepts of pointwise statistical convergence sequences of fuzzy mappings and established some basic properties of fuzzy mappings. Gong et al. [10] studied statistical convergence, uniformly statistical convergence and equi-statistical convergence for sequences of fuzzy valued functions and established some basic properties of sequences of fuzzy valued functions based on sequences of α-level cuts. Recently, Hazarika [12] introduced pointwise ideal convergence and uniformly ideal convergence of sequences of fuzzy valued functions.

In this article, we proposed the concept of ideal convergence, uniformly ideal convergence and equi-ideal convergence for sequences of fuzzy valued functions and proved some classical results in this new settings and their representations of sequences of α-level cuts. We proved the ideal version of the Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space $(X, M, μ) .$ Finally, we define the notion of ideal convergence in measure for sequences of fuzzy valued functions and prove some interesting results.

2 Preliminaries and definitions

In 1965, Zadeh [29] for the first time introduced the term Fuzzy sets for the class of objects with a continuum of grades of membership or in other words degrees of membership. He used this notion for the addition of inclusion, intersection, union, convexity, complement, relation, etc., in the classical notion sets and established the notion Fuzzy Sets. From then onward, fuzzy logic and fuzzy sets are successfully applied by the researchers of the various fields such as Decision theory, Control Engineering and also explored in Artificial Intelligence. The array of its application is continuously expanding since then. In the last 50 years, the sequences of fuzzy numbers and its convergence properties have been nicely demonstrated by many researchers. Let E be a nonempty set. According to Zadeh [29], a fuzzy subset of E is a nonempty subset ${(t, \bar{x} (t)) : t \in E}$ of E × J (= [0, 1]) for some function $\bar{x} : E \to J (= [0, 1]) .$ A function $\bar{x} : ℝ \to J (= [0, 1])$ is called a fuzzy number if the function $\bar{x}$ satisfies the following properties:

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

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

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

$[\bar{x}]^{0} = cl ({t \in ℝ : \bar{x} (t) > 0})$ is compact, where cl is the closure operator.

We denote the set of all fuzzy numbers by $F (R) .$ The set $ℝ$ of real numbers can be embedded in $F (R)$ if we define $\bar{r} \in F (ℝ)$ by $\bar{r} (t) = {\begin{matrix} 1, if t = r; \\ 0, if t \neq r . \end{matrix}$

For 0 < α ≤ 1, α-cut of $\bar{x}$ is defined by $[\bar{x}]_{α} = {t \in ℝ : \bar{x} (t) \geq α} = [{\bar{x}}_{α}^{-}, {\bar{x}}_{α}^{+}]$ is a closed and bounded interval of $ℝ .$ As in [18], the Hausdorff distance between two fuzzy numbers $\bar{x}$ and $\bar{y}$ given by $D : F (ℝ) \times F (ℝ) ⟶ [0, + \infty)$ $\begin{matrix} D (\bar{x}, \bar{y}) & = & sup_{α \in [0, 1]} d ([\bar{x}]_{α}, [\bar{y}]_{α}) \\ = & sup_{α \in [0, 1]} max {| {\bar{x}}_{α}^{-} - {\bar{x}}_{α}^{+} |, | {\bar{y}}_{α}^{-} - {\bar{y}}_{α}^{+} |}, \end{matrix}$ where d is the Hausdorff metric. For any $\bar{x}, \bar{y}, \bar{z}, \bar{u} \in F (ℝ)$ we know that

$(F (ℝ), D)$ is a complete metric space.

$D (γ \bar{x}, γ \bar{y}) = | γ | D (\bar{x}, \bar{y}); γ \in ℝ .$

$D (\bar{x} + \bar{u}, \bar{y} + \bar{u}) = D (\bar{x}, \bar{y}) .$

$D (\bar{x} + \bar{z}, \bar{y} + \bar{u}) \leq D (\bar{x}, \bar{y}) + D (\bar{z}, \bar{u}) .$

Lemma 2.1. [18] Let $\bar{x} \in F (ℝ)$ and $[\bar{x}]_{α} = [{\bar{x}}_{α}^{-}, {\bar{x}}_{α}^{+}] .$ Then the following conditions are satisfied:

${\bar{x}}_{α}^{-}$ is a left continuous monotone-nondecreasing function on (0, 1].

${\bar{x}}_{α}^{+}$ is a right continuous monotone-nonincreasing function on (0, 1].

${\bar{x}}_{α}^{-}$ and ${\bar{x}}_{α}^{+}$ are right continuous at α = 0.

${\bar{x}}_{1}^{-} \leq {\bar{x}}_{1}^{+} .$

Let M be a subset of $ℕ .$ The asymptotic density or density of M, denoted by δ (M), is given by $δ (M) = lim_{n \to \infty} \frac{1}{n} | {k \leq n : k \in M} |,$ if this limit exists, where | {k ≤ n : k ∈ M} | denotes the cardinality of the set {k ≤ n : k ∈ M}.

A sequence x = (x _n) is statistically convergent to ℓ if $δ ({n \in ℕ : | x_{n} - ℓ | \geq ε}) = 0$ for every ε > 0. In this case ℓ is called the statistical limit of x.

Let S be a non-empty set. Then a non empty class I ⊆ P (S) is said to be an ideal on S if and only if (i) φ ∈ I. (ii) I is additive under union (iii) hereditary. An ideal I ⊆ P (S) is said to be non trivial if I ≠ φ and S ∉ I. A non-empty family of sets F ⊆ P (S) is said to be a filter on S if and only if (i) φ ∉ F (ii) for each A, B ∈ F we have A ∩ B ∈ F (iii) for each A ∈ F and B ⊃ A, implies B ∈ F. For each ideal I, there is a filter F (I) corresponding to I i.e. F (I) = {K ⊆ S : K ^c ∈ I}, where K ^c = S - K. We say that a non-trivial ideal I ⊆ P (S) is (i) an admissible ideal on S if and only if it contains all singletons, i.e. if it contains {{x} : x ∈ S} (ii) maximal, if there cannot exists any non-trivial ideal J ≠ I containing I as a subset (iii) said to be a translation invariant ideal if {k + 1 : k ∈ A} ∈ I, for any A ∈ I. Recall that a sequence x = (x _k) of points in $ℝ$ is said to be I-convergent to the number ℓ (denoted by I - lim x _k =ℓ) if for every ε > 0, the set ${k \in ℕ, : | x_{k} - ℓ | \geq ε} \in I .$

Remark 2.1. If we take $I = I_{f} = {A \subseteq ℕ : A$ is a finite subset of $ℕ$ }, then I _f is a non-trivial admissible ideal of $ℕ$ and the corresponding convergence coincides with the usual convergence.

Remark 2.2. If we take $I = I_{δ} = {A \subseteq ℕ : δ (A) = 0}$ where δ (A) denote the asymptotic density of the set A, then I _δ is a non-trivial admissible ideal of $ℕ$ and the corresponding convergence coincides with the statistical convergence.

In this article we assume that I is an admissible ideal of $ℕ .$ We also assume that the ideals are proper $(\neq P (ℕ))$ and contain all finite sets. The ideals which consists of all finite sets is denoted by Fin. If I is an ideal and A ∉ I, then IupharpoonleftA, we mean the restriction of the ideal I to the set A, i.e. IupharpoonleftA = {B ⊂ A : B ∈ I}. An ideal I is a P-ideal if for every sequence $(A_{n})_{n \in ℕ}$ of sets from I there is an A ∈ I such that A _n \ A is finite for all n.

By identifying subsets of naturals with their characteristic functions, we equip $P (ℕ)$ with the Cantor space topology and therefore we can assign the topological complexity to the ideals of sets of naturals. In particular, an ideal I is analytic if it is a continuous image of a G _δ subset of the Cantor space. A nontrivial analytic P-deal is the ideal of sets of statistical density zero, i.e. $I_{δ} = {A \subset ℕ : \underset{j \to \infty}{lim sup} δ_{j} (A) = 0},$ where $δ_{j} (A) = \frac{| A \cap j |}{j}$ is the jth partial density of A. Another example of an analytic P-ideal is $\times ℕ .$ It is an ideal on $ℕ \times ℕ,$ which contains those subsets of $ℕ \times ℕ,$ whose all vertical sections are finite.

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

φ (varPhi) =0

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

It is lower semicontinuous if for all $A \subset ℕ$ we have $φ (A) = lim_{n \to \infty} φ (A \cap n) .$

For any lower semicontinuous submeasure on $ℕ,$ let $| | | . | | |_{φ} : P (ℕ) ⟶ [0, \infty)$ be the submeasure defined by $| | | A | | |_{φ} = \underset{n \to \infty}{lim sup} φ (A \ n) = lim_{n \to \infty} φ (A \ n),$ where the second equality follows by the monotonocity of φ. Let $Exh (φ) = {A \subset ℕ : | | | A | | |_{φ} = 0} .$

It is clear that Exh (φ) is an ideal (not necessarily proper) for an arbitrary submeasure φ.

Lemma 2.2. [27] The following conditions are equivalent for an ideal I on $ℕ .$

I is an analytic P-ideal

I = Exh (φ) for some lower semicontinuous submeasure φ on $ℕ .$

Lemma 2.3. [17] Suppose that I = Exh (φ) for some lower semicontinuous submeasure. Then I is dense if and only if $lim_{n \to \infty} φ ({n}) = 0 .$

Lemma 2.4. [17] If I is isomorphic to Fin or varPhi × Fin then $lim_{n \in A} φ ({n}) = 0$ for some A ∉ I.

3 Ideal convergence of sequences of fuzzy valued functions

In this section we define pointwise ideal convergence, uniformly ideal convergence and equi-ideal convergence for sequence of fuzzy valued functions defined on [a, b]. Suppose that $\bar{f} : [a, b] ⟶ F (ℝ)$ is a fuzzy valued function and ${\bar{f}}_{n} : [a, b] ⟶ F (ℝ), n \in ℕ$ is a sequence of fuzzy valued functions.

Gong et al. [10] introduced the notion of equi-statistical convergence of fuzzy valued functions. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is equi-statistical convergent to $\bar{f}$ ( ${\bar{f}}_{n} eS \bar{f}$ ) if and only if $\begin{matrix} \forall ε, σ > 0, \exists k \in ℕ, \forall j \geq k, \forall x \in X, \\ δ_{j} ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε}) < σ . \end{matrix}$ Definition 3.1. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to be pointwise ideally convergent to a fuzzy valued function $\bar{f}$ on [a, b], if ${\bar{f}}_{n} (x) I ⟶ \bar{f} (x)$ for each x ∈ [a, b], i.e. for every ε > 0, σ > 0, for each x ∈ [a, b] $\begin{matrix} {k \in ℕ : φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε} \ k) \\ \geq σ} \in I . \end{matrix}$

This can be written as $\begin{matrix} \forall x \in [a, b], \forall ε > 0, \forall σ > 0 \exists M_{x} \in I such that \\ φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε} \ k) < σ, \\ \forall k \in ℕ \ M_{x} . \end{matrix}$

We write ${\bar{f}}_{n} pI ⟶ \bar{f}$ on [a, b].

Definition 3.2. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to be uniformly ideally convergent to a fuzzy valued function $\bar{f}$ on [a, b], if for every ε > 0, σ > 0, ∀x ∈ [a, b] $\begin{matrix} {k \in ℕ : φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \\ \geq ε} \ k) \geq σ} \in I . \end{matrix}$

This can be written as $\begin{matrix} \forall x \in [a, b], \forall ε > 0, \forall σ > 0 \exists M \in I such that \\ φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε} \ k) < σ, \\ \forall k \in ℕ \ M . \end{matrix}$

We write ${\bar{f}}_{n} uI ⇉ \bar{f}$ on [a, b].

Example 3.1. For any x ∈ [0, 1] we define ${\bar{f}}_{n} (x) = {\begin{matrix} \bar{(\frac{nx}{1 + n^{2} x^{2}})}, & for x \in [0, 1]; \\ \bar{0} & otherwise . \end{matrix}$

Then $({\bar{f}}_{n})$ pointwise ideal converges to $\bar{f} = \bar{0} .$ But for all $n \in ℕ$ $\begin{matrix} sup_{x \in [0, 1]} D ({\bar{f}}_{n} (x), \bar{0}) = sup_{x \in [0, 1]} \frac{nx}{1 + n^{2} x^{2}} \\ = sup_{x \in [0, 1]} \frac{1}{\frac{1}{nx} + nx} = \frac{1}{2} \neq 0 . \end{matrix}$

Hence $({\bar{f}}_{n})$ is not uniformly convergent and uniformly ideal convergent to $\bar{0}$ on [0, 1].

Definition 3.3. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is called equi-ideally convergent to a fuzzy valued function $\bar{f}$ if $\begin{matrix} \forall ε > 0, \forall σ > 0 such that \\ {k \in ℕ : φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \\ \geq ε} \ k) \geq σ} \end{matrix}$ with respect to x ∈ [a, b] is uniformly convergent to zero function. In this case we write ${\bar{f}}_{n} eI \bar{f} .$

Example 3.2. For any x ∈ [0, 1] we define $\begin{matrix} {\bar{g}}_{n} (x) \\ = {\begin{matrix} \bar{2^{n + 1} (x - \frac{1}{2^{n}})}, & for x \in [\frac{1}{2^{n}}, \frac{1}{2^{n - 1}} - \frac{1}{2^{n + 1}}]; \\ \bar{- 2^{n + 1} (x - \frac{1}{2^{n - 1}})}, & for x \in [\frac{1}{2^{n - 1}} - \frac{1}{2^{n + 1}}, \frac{1}{2^{n - 1}}]; \\ \bar{0} & otherwise . \end{matrix} \end{matrix}$

Then $({\bar{g}}_{n})$ equi-ideal converges to a function $\bar{g} = \bar{0} .$ But $({\bar{g}}_{n})$ is not uniformly ideal convergent and uniform convergent to $\bar{0}$ on [0, 1].

Remark 3.1. ${\bar{f}}_{n} eI \bar{f}$ if and only if $\begin{matrix} \forall ε > 0, \forall σ > 0 \exists k \in ℕ, \forall x \in [a, b] such that \\ φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε} \ k) < σ . \end{matrix}$

Remark 3.2. ${\bar{f}}_{n} uI ⇉ \bar{f} \Rightarrow {\bar{f}}_{n} eI \bar{f} \Rightarrow {\bar{f}}_{n} pI ⟶ \bar{f}$

Remark 3.3. $\neg ({\bar{f}}_{n} eI \bar{f}) \Leftarrow {\bar{f}}_{n} pI ⟶ \bar{f} .$ For any ideal I, we can use an example where ${\bar{f}}_{n}$ is a characteristic function of $[0, \frac{1}{n}]$ for $n \in ℕ .$

Proposition 3.1. Equi-ideal convergence is well defined with respect to α for α ∈ [0, 1].

Proof. For the sake of contradiction, suppose that there exist two lower semicontinuous submeasures φ ₁ and φ ₂ such that Exh (φ ₁) = Exh (φ ₂) and $\begin{matrix} \forall ε > 0, \exists k \in ℕ, \forall x \in X, \\ φ_{1} ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε} \ k) < ε; \\ \forall \bar{ε} > 0, \exists k \in ℕ, \forall x \in X, \\ φ_{2} ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \bar{ε}} \ k) < \bar{ε} . \end{matrix}$

For all x ∈ X and for α ∈ [0, 1] we denote $\begin{matrix} U (x) & = & {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \bar{ε}} \\ = & {n \in ℕ : sup_{α \in [0, 1]} max {| ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, \\ | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} \geq \bar{ε}} . \end{matrix}$ and $\begin{matrix} U_{1} (x) & = & {n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \geq \bar{ε}}, \\ U_{2} (x) & = & {n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \geq \bar{ε}} . \end{matrix}$

We find k ₁ and x ₁ such that $\begin{matrix} φ_{1} (U_{1} (x_{1}) \ k_{1}) < \frac{1}{2}, φ_{1} (U_{2} (x_{1}) \ k_{1}) < \frac{1}{2} \\ and φ_{2} (U_{1} (x_{1}) \ k_{1}) \geq \bar{ε}, φ_{2} (U_{2} (x_{1}) \ k_{1}) \geq \bar{ε} . \end{matrix}$

Since submeasure φ ₂ is lower continuous, we can find with $\begin{array}{l} φ_{2} & ((U_{1} (x_{1}) \ k_{1}) \cap {k^{'}}_{1}) \\ > \frac{\bar{ε}}{2}, φ_{2} ((U_{1} (x_{1}) \ k_{1}) \cap {k^{'}}_{1}) > \frac{\bar{ε}}{2} . \end{array}$

Suppose that we have already found Let $k_{i + 1} > k i'$ and x _i+1 such that $\begin{matrix} φ_{1} (U_{1} (x_{i + 1}) \ k_{i + 1}) < \frac{1}{2^{i + 1}}, \\ φ_{1} (U_{2} (x_{i + 1}) \ k_{i + 1}) < \frac{1}{2^{i + 1}} \\ and φ_{2} (U_{1} (x_{i + 1}) \ k_{i + 1}) \geq \bar{ε}, φ_{2} (U_{2} (x_{i + 1}) \\ \ k_{i + 1}) \geq \bar{ε} . \end{matrix}$

Again by the lower continuity of φ ₂ we find ${k^{'}}_{i + 1}$ with $\begin{array}{l} φ_{2} ((U_{1} (x_{i + 1}) \ k_{1}) \cap {k^{'}}_{i + 1}) > \frac{\bar{ε}}{2}, \\ φ_{2} ((U_{1} (x_{x_{i + 1}}) \ k_{i +}_{1}) \cap {k^{'}}_{i + 1}) > \frac{\bar{ε}}{2} . \end{array}$

We put $\begin{array}{l} A = \underset{i \in ℕ}{\cup} ((U_{1} (x_{i}) \ k_{i}) \cap {k^{'}}_{i}) and \\ A = \underset{i \in ℕ}{\cup} ((U_{1} (x_{i}) \ k_{i}) \cap {k^{'}}_{i}) . \end{array}$

Then we have $\begin{matrix} | | A | |_{φ_{1}} & = & \lim_{i \to \infty} φ_{1} (A \ k_{i}) \\ \leq & \lim_{i \to \infty} \sum_{n = i}^{\infty} φ_{1} (U_{1} (x_{i}) \cap^{} {k_{i}, k_{i}^{'}}) \\ \leq & \lim_{i \to \infty} \sum_{n = i}^{\infty} \frac{1}{2^{n}} = 0 \end{matrix}$ and $\begin{matrix} | | B | |_{φ_{1}} & = & \lim_{i \to \infty} φ_{1} (B \ k_{i}) \\ \leq & \lim_{i \to \infty} \sum_{n = i}^{\infty} φ_{1} (U_{2} (x_{i}) \cap^{} {k_{i}, k_{i}^{'}}) \\ \leq & \lim_{i \to \infty} \sum_{n = i}^{\infty} \frac{1}{2^{n}} = 0. \end{matrix}$

Therefore we get A ∈ Exh (φ ₁) and B ∈ Exh (φ ₁).

On the other hand we have $\begin{matrix} | | A | |_{φ_{2}} = lim_{i \to \infty} φ_{2} (A \ k_{i}) \geq \frac{\bar{ε}}{2} > 0 \\ and | | B | |_{φ_{2}} = lim_{i \to \infty} φ_{2} (B \ k_{i}) \geq \frac{\bar{ε}}{2} > 0 \end{matrix}$ which is a contradiction. This contradicts imply the proof of the theorem.

Proposition 3.2. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is equi-statistical convergent to a fuzzy valued function $\bar{f}$ if and only if it is equi-ideal convergent to $\bar{f}$ with respect to the ideal I _δ.

Proof. We have I _δ = Exh (φ), where $φ (A) = sup_{j} δ_{j} (A)$ for $A \subset ℕ .$ Therefore we have the followings:

if δ _j (A) < ε for j ≥ k then $sup_{j} δ_{j} (A \ k) \leq ε$

if $sup_{j} δ_{j} (A \ k) \leq ε$ then δ _j (A) ≤ δ _j (A \ k) + δ _j (k) ≤2ε for $j \geq \frac{k}{ε} .$

These two results completes the proof of the theorem.

Proposition 3.3. Let $({\bar{f}}_{n})$ be sequence of fuzzy valued functions and $\bar{f}$ be a fuzzy valued function. Then ${\bar{f}}_{n} e Fin \bar{f} \Rightarrow {\bar{f}}_{n} e Fin ⇉ \bar{f} .$

Proof. We have Fin = Exh (φ) where $φ (A) = {\begin{matrix} | A |, if A is finite; \\ \infty, if A is infinite \end{matrix}$

Suppose that ${\bar{f}}_{n} e Fin \bar{f} .$ Take 0 < ε < 1. There exists $n \in ℕ$ such that $\forall x \in X, φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε} \ k) < ε .$

Since φ admits only integer values, therefore we have $φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε} \ k) = 0 .$

This implies that this set is empty for each x ∈ X. Hence ${\bar{f}}_{n} e Fin ⇉ \bar{f} .$

Proposition 3.4. Suppose that I = Exh (φ) for some lower semicontinuous submeasure. Then I is isomorphic to Fin or varPhi × Fin if and only if for any sequences of fuzzy valued functions equi-ideal convergence and uniform ideal convergence are equivalent.

Proof. Suppose an ideal I = varPhi × Fin and suppose that ${\bar{f}}_{n} eI \bar{f} .$ To show that ${\bar{f}}_{n} uI ⇉ \bar{f} .$ If possible suppose that $\neg ({\bar{f}}_{n} uI ⇉ \bar{f}) .$ Then for some ε > 0 and for all x ∈ X, we have $A = {n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{2}} \notin I .$

There exists B ⊂ A, B ∉ I such that I ↾ B = Fin. Therefore we have ${\bar{f}}_{n} eI ↾ B \bar{f}$ and $\neg ({\bar{f}}_{n} uI ↾ B ⇉ \bar{f})$ which contradicts the Proposition 3.3.

Next we take any analytic P-ideal I = varPhi × Fin which is not isomorphic to Fin or varPhi × Fin. By the Lemma 2.4, we have $lim_{n \in A} φ ({n}) = 0$ for some A ∉ I. Let ${\bar{f}}_{n}$ be the characteristic function of ${\frac{1}{n}}$ for n ∈ A and zero function otherwise. Then for each x ∈ X, we have ${n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{2}} = A \notin I$ so, $\neg ({\bar{f}}_{n} uI ⇉ \bar{f}) .$

We show that ${\bar{f}}_{n} eI \bar{f} .$ Let ε > 0 be given and $m \in ℕ$ such that φ ({n}) < ε for n > m. For any α ∈ [0, 1], the set $B_{x} = {n \in ℕ : sup_{α \in [0, 1]} max {| ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} \geq ε}$ contains at most one element for each x ∈ [0, 1]. Hence μ (B _x \ m) < ε and so ${\bar{f}}_{n} eI \bar{f} .$

Proposition 3.5. If I is a translation invariant ideal and $I - lim_{k \to \infty} {\bar{f}}_{k} (x) = \bar{f} (x),$ then $I - lim_{k \to \infty} {\bar{f}}_{k + 1} (x) = \bar{f} (x)$ for each x ∈ [a, b].

Proof. The proof of the result is easy, so omitted.

Theorem 3.1. Let I be an admissible ideal. Let $({\bar{f}}_{n})$ be a sequence of fuzzy valued functions and $\bar{f}$ be a fuzzy valued function defined on [a, b]. For each x ∈ [a, b], if ${\bar{f}}_{n} (x) p ⟶ \bar{f} (x)$ then $[{\bar{f}}_{n} (x)]_{α} pI ⟶ [\bar{f} (x)]_{α}$ with respect to α.

Proof. Suppose that for each x ∈ [a, b], ${\bar{f}}_{n} (x) p ⟶ \bar{f} (x) .$ Let ε > 0 be given. For each x ∈ X, then there exists an integer $m = m (x, ε) \in ℕ$ such that $\begin{matrix} \forall ε > 0, \forall σ > 0, \\ φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε} \ k) < σ \end{matrix}$ (1) $for all k \geq m .$

This implies that for any α ∈ [0, 1], ∀ε > 0, ∀ σ > 0, $\begin{matrix} {n \in ℕ : sup_{α \in} [0, 1] max {| ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} |, \\ | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} \geq ε} \ k) < σ \end{matrix}$ (2) $for all k \geq m .$

For any α ∈ [0, 1] we have $\begin{matrix} \forall ε > 0, \forall σ > 0, \\ φ ({n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \geq ε} \ k) \end{matrix}$ (3) $< σ for all k \geq m .$ and $\begin{matrix} \forall ε > 0, \forall σ > 0, \\ φ ({n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \geq ε} \ k) \end{matrix}$ (4) $< σ for all k \geq m .$

From relation (3.1) we have $\begin{matrix} \forall ε > 0, \forall σ > 0, \\ {k \in ℕ : φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq ε} \ k) \geq σ} \\ \subset ℕ \cup {1, 2, . . ., m - 1} . \end{matrix}$

Again from (3.2) we get $\begin{matrix} \forall ε > 0, \forall σ > 0, \\ {k \in ℕ : φ ({n \in ℕ : sup_{α \in [0, 1]} max {| ({\bar{f}}_{n} (x))_{α}^{-} \\ - (\bar{f} (x))_{α}^{-} |, \\ | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} |} \geq ε} \ k) \geq σ} \\ \subset ℕ \cup {1, 2, . . ., m - 1} . \end{matrix}$

For any α ∈ [0, 1]. Then from (3.3) and (3.4) we get $\begin{matrix} \forall ε > 0, \forall σ > 0, \\ {k \in ℕ : φ ({n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \\ \geq ε} \ k) \geq σ} \\ \subset ℕ \cup {1, 2, . . ., m - 1} \end{matrix}$ and $\begin{matrix} \forall ε > 0, \forall σ > 0, \\ {k \in ℕ : φ ({n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \\ \geq ε} \ k) \geq σ} \\ \subset ℕ \cup {1, 2, . . ., m - 1} . \end{matrix}$

Since I is an admissible ideal, therefore we have $ℕ \cup {1, 2, . . ., m - 1} \in I$ and consequently we have $\begin{matrix} \forall ε > 0, \forall σ > 0, \\ {k \in ℕ : φ ({n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{-} - (\bar{f} (x))_{α}^{-} | \\ \geq ε} \ k) \geq σ} \in I \end{matrix}$ and $\begin{matrix} \forall ε > 0, \forall σ > 0, \\ {k \in ℕ : φ ({n \in ℕ : | ({\bar{f}}_{n} (x))_{α}^{+} - (\bar{f} (x))_{α}^{+} | \\ \geq ε} \ k) \geq σ} \in I . \end{matrix}$

Therefore for each x ∈ [a, b] we get $[{\bar{f}}_{n} (x)]_{α} pI ⟶ [\bar{f} (x)]_{α}$ with respect to α.

4 Egorov’s theorem for the sequence of fuzzy valued functions in ideal context

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

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

Theorem 4.1. Let I be an analytic P-ideal and let $(X, M, μ)$ be a finite measure space. Let the fuzzy valued function $\bar{f}$ and the sequence of fuzzy valued function $({\bar{f}}_{n})$ be measurable and defined on almost everywhere on X. Let $({\bar{f}}_{n})$ be pointwise ideally convergent to a fuzzy valued function $\bar{f}$ almost everywhere on X. Then for every ε > 0 there exists a subset A of X such that μ (X \ A) < ε and $({\bar{f}}_{n | A})$ is equi-ideally convergent to ${\bar{f}}_{| A}$ on A.

Proof. Let $(X, M, μ)$ be a finite measure space, $\bar{f}, {\bar{f}}_{n} : X ⟶ F (ℝ)$ for each $n \in ℕ .$ Without loss of generality we assume that for all $n \in ℕ, \bar{f}, {\bar{f}}_{n}$ are defined everywhere on X and $({\bar{f}}_{n})$ be pointwise ideally convergent to a fuzzy valued function $\bar{f}$ almost everywhere on X.

For fix $k, q \in ℕ$ we define the set $\begin{matrix} E_{k, q} = {x \in X : φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \\ \geq \frac{1}{q}} \ k) \leq \frac{1}{q}} . \end{matrix}$

We show that the sets E _k,q is measurable. For this we show that the complement of each E _k,q is measurable. $\begin{matrix} X \ E_{k, q} \\ = {x \in X : ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{q}} \ k) \\ \in φ^{- 1} ((\frac{1}{q}, \infty))} . \end{matrix}$

Since φ is lower semicontinuous, there exist sets [A _i, n _i] such that $\begin{matrix} X \ E_{k, q} \\ = {x \in X : ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{q}} \ k) \\ \in ⋃_{i \in ℕ} [A_{i}, n_{i}]} \\ = ⋃_{i \in ℕ} ⋂_{j = k}^{n_{i}} {x \in X : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{q}}^{χ_{A_{i} (j)}} . \end{matrix}$

Since $\bar{f}, {\bar{f}}_{n}$ are measurable for each $n \in ℕ,$ therefore the right hand side set in the above relation is measurable and hence the set X \ E _k,q is measurable. Now for each $q \in ℕ$ we have E _k,q ⊂ E _k+1,q and $X = ⋃_{k = 1}^{\infty} E_{k, q} .$ Consequently $μ (X) = lim_{n \to \infty} μ (E_{k, q}) .$ Let ε > 0 be given. For each $q \in ℕ,$ let $k (q) \in ℕ$ be such that $μ (X \ E_{k (q), q}) < \frac{ε}{2^{q}} .$ We put $A_{0} = ⋃_{q = 1}^{\infty} (X \ E_{k (q), q}) .$ Then we get $μ (A_{0}) \leq \sum_{q = 1}^{\infty} μ (X \ E_{k (q), q}) < ε .$ Let $A = X \ A_{0} = ⋂_{q = 1}^{\infty} E_{k (q), q} .$ Then μ (X \ A) = μ (A ₀) < ε. Therefore we have $\begin{matrix} \forall q \in ℕ, \exists k (q) \in ℕ, \forall x \in A, \\ φ ({n \in ℕ : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq \frac{1}{q}} \ k) < \frac{1}{q} . \end{matrix}$

This shows that $({\bar{f}}_{n | A})$ is equi-ideally convergent to ${\bar{f}}_{| A}$ on A.

Corollary 4.1. Let $(X, M, μ)$ be a finite measure space. Let fuzzy valued function $\bar{f}$ and the sequence of fuzzy valued functions $({\bar{f}}_{n})$ be measurable and defined almost everywhere on X. Then $({\bar{f}}_{n})$ is ideally convergent to $\bar{f}$ almost everywhere on X if and only if there is a sequence (A _k) of sets on X such that ${\bar{f}}_{n | A_{k}} eI ⟶ {\bar{f}}_{| A_{k}}$ on A _k for all k and $μ (X \ ⋃_{k \in ℕ} A_{k}) = 0 .$

Proof. For the part “ ⇒ ″, consider $ε = \frac{1}{k}, k \in ℕ$ in the Theorem 4.1.

The part “ ⇐ ″ follows from ${\bar{f}}_{n | A_{k}} eI ⟶ {\bar{f}}_{| A_{k}} \Rightarrow {\bar{f}}_{n | A_{k}} I ⟶ {\bar{f}}_{| A_{k}}$ for $k \in ℕ .$

5 Ideal convergence in measure of sequences of fuzzy valued functions

In this section we define the ideal convergence in measure for a sequence of fuzzy valued functions and obtain some results using this notion.

Definition 5.1. Let $({\bar{f}}_{n}) \in L^{0}, \bar{f} \in L^{0}$ and X be a finite measurable set. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to be convergent in measure to a fuzzy valued function $\bar{f}$ if $μ ({x \in X : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq η})$ is convergent to $\bar{0}$ for every η > 0 and all $n \in ℕ .$ We write ${\bar{f}}_{n} μ ⟶ \bar{f} .$

Definition 5.2. Let $({\bar{f}}_{n}) \in L^{0}, \bar{f} \in L^{0}$ and X be a finite measurable set. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is said to be ideally convergent in measure to a fuzzy valued function $\bar{f}$ if $μ ({x \in X : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq η})$ is ideally convergent to zero for every η > 0 and all $n \in ℕ .$ We write ${\bar{f}}_{n} μ I ⟶ \bar{f} .$ It is equivalent to the condition $\begin{matrix} \forall r > 0, \forall η > 0, \\ {n \in ℕ : μ ({x \in X : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq η}) \geq r} \in I, \end{matrix}$ in which we may take η = r or $η = \frac{1}{q}, q \in ℕ .$

Remark 5.1. We have ${\bar{f}}_{n} μ ⟶ \bar{f} \Rightarrow {\bar{f}}_{n} μ I ⟶ \bar{f},$ but the converse need not be true. It will be followed from the following example.

Example 5.1. Let the sequence of fuzzy valued functions $({\bar{f}}_{n})$ be defined by ${\bar{f}}_{1}^{(1)} (x) (t) = {\begin{matrix} g (x) (t) & for x \in [0, \frac{1}{2}); \\ \bar{0} & for x \in [\frac{1}{2}, 1), \end{matrix}$ $g (x) (t) = {\begin{matrix} 0 & for t \in (- \infty, 0) \cup (2, + \infty); \\ t - 1 & for t \in [0, 1] \\ 2 - t & for t \in (1, 2]; \end{matrix}$ ${\bar{f}}_{2}^{(1)} (x) (t) = {\begin{matrix} \bar{0} & for x \in [0, \frac{1}{2}); \\ g (x) (t) & for x \in [\frac{1}{2}, 1), \end{matrix}$

We divide the interval [0, 1) into four equal parts, eight equal parts, and so on. In this way for every n we get the following functions: ${\bar{f}}_{j}^{(n)} (x) (t) = {\begin{matrix} g (x) (t) & for x \in [\frac{j - 1}{2^{n}}, \frac{j}{2^{n}}); \\ \bar{0} & for x \in [\frac{j - 1}{2^{n}}, \frac{j}{2^{n}}), \end{matrix}$ for j = 1, 2, 2²,…, 2ⁿ.

Now arrange ${{\bar{f}}_{j}^{(n)}, j = 1, 2, 2^{2}, . . ., 2^{n}}$ as follows: ${\bar{f}}_{1}^{(1)}, {\bar{f}}_{2}^{(1)}, . . ., {\bar{f}}_{1}^{(n)}, {\bar{f}}_{2}^{(n)}, . . ., {\bar{f}}_{2^{n}}^{(n)}, . . .$

Let the sequence of fuzzy valued functions $({\bar{g}}_{n} (x))$ consist of the same functions ordered on $({\bar{f}}_{j}^{(n)} (x))$ but each function $({\bar{f}}_{j}^{(n)} (x))$ is repeated 2^n-1 times. Then the sequence of fuzzy valued functions $({\bar{g}}_{n} (x))$ is ideally convergent in measure to $\bar{0}$ on [0, 1). There is no point on [0, 1) such that a sequence of fuzzy valued functions $({\bar{g}}_{n} (x))$ is ideally convergent to $\bar{0}$ on [0, 1). Then for any η > 0 we have $\begin{matrix} μ ({x \in [0, 1] : D ({\bar{g}}_{n} (x), \bar{0}) \geq η}) = 0 \\ or μ ({x \in [0, 1] : D ({\bar{g}}_{n} (x), \bar{0}) \geq η}) \\ = \frac{j}{2^{n}} - \frac{j - 1}{2^{n}} = \frac{1}{2^{n}} < \frac{1}{n} . \end{matrix}$

Let ε > 0 and take $m = \frac{1}{ε} + 1 .$ If n > m then $μ ({x \in [0, 1] : D ({\bar{g}}_{n} (x), \bar{0}) \geq η})$ is ideally convergent to $\bar{0} .$ Therefore it is convergent to $\bar{0}$ in usual sense. By the definition of ideal convergence in measure for the sequence of fuzzy valued functions, the sequence $({\bar{g}}_{n})$ of fuzzy valued functions is ideally convergent in measure to $\bar{0} .$

Otherwise, let $ε = \frac{1}{2}$ and define $j_{n} = \sum_{j = 1}^{n} 2^{j - 1} = 2^{n - 1}$ for $n \in ℕ .$ If x ∈ [0, 1] there exists an increasing subsequence (j _{n
_k}) of (j _n) for $k \in ℕ$ such that $\begin{matrix} δ_{j_{n_{k}}} ({m \in ℕ : g_{m} (x) = \bar{1}}) \geq \frac{2^{n_{k} - 1}}{2^{n_{k}} - 1} \\ \geq \frac{1}{2} = ε for k \in ℕ . \end{matrix}$

Therefore, for I = I _δ, (g _n (x)) is not I _δ-convergent to $\bar{0} .$

Proposition 5.1. Let $({\bar{f}}_{n}) \in L^{0}, \bar{f} \in L^{0}$ and X be a finite measurable set. Then ${\bar{f}}_{n} I ⇉ \bar{f} \Rightarrow {\bar{f}}_{n} μ I ⟶ \bar{f} .$

Proof. We suppose that ${\bar{f}}_{n} I ⇉ \bar{f} .$ and let η > 0. Then there is a set M ∈ I such that $D ({\bar{f}}_{n} (x), \bar{f} (x)) < η$ for all n ∉ M and x ∈ X. Thus we have $\begin{matrix} {n \in ℕ : μ ({x \in X : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq η}) \geq η} \\ \subset {n \in ℕ : μ ({x \in X : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq η}) \neq φ} \\ \subset M \in I . \end{matrix}$

This shows that ${\bar{f}}_{n} (x) μ I ⟶ \bar{f} (x) .$

Theorem 5.1. Let $({\bar{f}}_{n}) \in L^{0},$ and X be a finite measurable set. Let $\bar{f}$ be a fuzzy valued measurable function such that for each x ∈ X, ${\bar{f}}_{n} (x) ⟶ \bar{f} (x) .$ Then given ε > 0 and δ > 0, there is a measurable set A ⊂ X with μ (A) < δ and an integer m such that $D ({\bar{f}}_{n} (x), \bar{f (x)}) < ε for all x \notin A and for all n \geq m .$

Proof. Let $G_{n} = {x \in X : D ({\bar{f}}_{n} (x), \bar{f (x)}) \geq ε}$ and set $\begin{matrix} X_{m} & = & ⋃_{n = m}^{\infty} {x \in X : D ({\bar{f}}_{n} (x), \bar{f (x)}) \\ \geq ε for some n \geq m} . \end{matrix}$

We have X _m+1 ⊂ X _m, and for each x ∈ X there must some X _n to which x does not belong, since ${\bar{f}}_{n} (x) ⟶ \bar{f} (x) .$ Thus we have ⋂X _m = φ, and so we have lim μ (X _m) =0. Hence given δ > 0, ∃ m so that μ (X _m)< δ ; i.e. $\begin{matrix} μ ({x \in X : D ({\bar{f}}_{n} (x), \bar{f (x)}) \geq ε \\ for some n \geq m}) < δ . \end{matrix}$

If we write A for this X _m, then μ (A) < δ and $\tilde{A} = {x \in X : D ({\bar{f}}_{n} (x), \bar{f (x)}) \geq ε for some n \geq m} .$

Corollary 5.1. Let $({\bar{f}}_{n}), \bar{f} \in L^{0},$ and X be a finite measurable set. If $({\bar{f}}_{n})$ converge to $\bar{f}$ a.e. on X. Then given ε > 0 and δ > 0, there is a measurable set A ⊂ X with μ (A) < δ and an integer m such that $D ({\bar{f}}_{n} (x), \bar{f} (x)) < ε for all x \notin A and for all n \geq m .$

Theorem 5.2. Let $({\bar{f}}_{n}), \bar{f} \in L^{0},$ and X be a finite measurable set. A sequence of fuzzy valued functions $({\bar{f}}_{n})$ is ideally convergent in measure to $\bar{f}$ if and only if $[{\bar{f}}_{n} (x)]_{α}$ is ideally convergent in measure to $[\bar{f} (x)]_{α}$ with respect to α.

Proof. Suppose that $({\bar{f}}_{n})$ is ideally convergent in measure to $\bar{f} .$ Then we have $μ ({x \in X : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq η})$ is ideally convergent to zero for every η > 0 and all $n \in ℕ .$ i.e. ${n \in ℕ : μ ({x \in X : D ({\bar{f}}_{n} (x), \bar{f} (x)) \geq η}) \geq η} \in I .$

Therefore we have $\begin{matrix} {n \in ℕ : μ ({x \in X : \\ sup_{α \in [0, 1]} max {| (f_{n} (x))_{α}^{-} - (f (x))_{α}^{-} |, \\ | (f_{n} (x))_{α}^{+} - (f (x))_{α}^{+} |} \geq η}) \geq η} \in I . \end{matrix}$

Thus for α ∈ [0, 1] we have $\begin{matrix} {n \in ℕ : μ ({x \in X : | (f_{n} (x))_{α}^{-} - (f (x))_{α}^{-} | \\ \geq η}) \geq η} \in I \end{matrix}$ and $\begin{matrix} {n \in ℕ : μ ({x \in X : | (f_{n} (x))_{α}^{+} - (f (x))_{α}^{+} | \\ \geq η}) \geq η} \in I . \end{matrix}$

This means that for any α ∈ [0, 1] we have $μ ({x \in X : | (f_{n} (x))_{α}^{-} - (f (x))_{α}^{-} | \geq η}) μ I ⟶ 0$ and $μ ({x \in X : | (f_{n} (x))_{α}^{+} - (f (x))_{α}^{+} | \geq η}) μ I ⟶ 0 .$

Hence $[{\bar{f}}_{n} (x)]_{α}$ is ideally convergent in measure to $[\bar{f} (x)]_{α}$ with respect to α.

Next we suppose that $[{\bar{f}}_{n} (x)]_{α}$ is ideally convergent in measure to $[\bar{f} (x)]_{α}$ with respect to α. Let η > 0. Then we have $μ ({x \in X : | (f_{n} (x))_{α}^{-} - (f (x))_{α}^{-} | \geq η}) μ I ⟶ 0$ and $μ ({x \in X : | (f_{n} (x))_{α}^{+} - (f (x))_{α}^{+} | \geq η}) μ I ⟶ 0 .$

Then for any α ∈ [0, 1] we get $\begin{matrix} {n \in ℕ : μ ({x \in X : | (f_{n} (x))_{α}^{-} - (f (x))_{α}^{-} | \\ \geq η}) \geq η} \in I \end{matrix}$ and $\begin{matrix} {n \in ℕ : μ ({x \in X : | (f_{n} (x))_{α}^{+} - (f (x))_{α}^{+} | \\ \geq η}) \geq η} \in I . \end{matrix}$

Therefore we get $\begin{matrix} {n \in ℕ : μ ({x \in X : \\ sup_{α \in [0, 1]} max {| (f_{n} (x))_{α}^{-} - (f (x))_{α}^{-} |, \\ | (f_{n} (x))_{α}^{+} - (f (x))_{α}^{+} |} \geq η}) \geq η} \in I . \end{matrix}$ $\begin{matrix} i . e {n \in ℕ : μ ({x \in X : D ({\bar{f}}_{n} (x), \bar{f} (x)) \\ \geq η}) \geq η} \in I . \end{matrix}$

This shows that $({\bar{f}}_{n})$ is ideally convergent in measure to $\bar{f} .$

Corollary 5.2. Let $({\bar{f}}_{n}), \bar{f} \in L^{0},$ and X be a finite measurable set. If ${\bar{f}}_{n} (x) μ I ⟶ \bar{f} (x),$ then there exists a subsequence $({\bar{f}}_{n_{k}})$ of $({\bar{f}}_{n})$ which is ideally convergent to $\bar{f}$ almost everywhere on X.

Definition 5.3. Let $({\bar{f}}_{n})$ be a sequence of fuzzy valued measurable functions in $L^{0} .$ Then $({\bar{f}}_{n})$ is said to be Cauchy in measure if $\forall ε > 0, μ (D ({\bar{f}}_{n}, {\bar{f}}_{m}) \geq ε) ⟶ 0, as n, m ⟶ \infty .$

This means that $\begin{matrix} \forall ε, η > 0, \exists N > 0 such that \\ n, m > N \Rightarrow μ (D ({\bar{f}}_{n}, {\bar{f}}_{m}) \geq ε) < η . \end{matrix}$

Definition 5.4. Let $({\bar{f}}_{n})$ be a sequence of fuzzy valued measurable functions in $L^{0} .$ Then $({\bar{f}}_{n})$ is said to be ideally Cauchy in measure if there exists an integer $m \in ℕ$ such that $\begin{matrix} \forall ε > 0, \forall η > 0, {n \in ℕ : μ (D ({\bar{f}}_{n}, {\bar{f}}_{m})) \\ \geq ε) \geq η} \in I . \end{matrix}$

Remark 5.2. Let $({\bar{f}}_{n})$ be a sequence of fuzzy valued measurable functions in $L^{0} .$ Then we have

$E = {x \in [a, b] : I - lim_{n ⟶ \infty} {\bar{f}}_{n} (x) exists}$ is measurable.

$\bar{f} (x) = {\begin{matrix} I - lim_{n ⟶ \infty} \bar{f} (x) & if the limit exists, \\ \bar{0} & otherwise . \end{matrix}$ is a measurable function.

Theorem 5.3. Let $({\bar{f}}_{n})$ be a sequence of fuzzy valued measurable functions in $L^{0} .$ Then the followings are equivalent.

$({\bar{f}}_{n})$ is ideally Cauchy in measure.

there exists an $\bar{f} \in L^{0}$ such that ${\bar{f}}_{n} μ I ⟶ \bar{f} .$

Proof. (a) ⇒ (b) Suppose that $({\bar{f}}_{n})$ is ideally Cauchy in measure. We show that there exists a subsequence $({\bar{g}}_{i})_{i \in ℕ}$ of $({\bar{f}}_{n})$ such that ${i \in ℕ : μ (D ({\bar{g}}_{i}, {\bar{g}}_{i + 1})) \geq \frac{1}{2^{i}}) \leq \frac{1}{2^{i}}} \in F .$

Define $\begin{matrix} E_{i} & = & {i \in ℕ : μ (D ({\bar{g}}_{i}, {\bar{g}}_{i + 1})) \geq \frac{1}{2^{i}}} \\ and H_{k} & = & ⋃_{j = k}^{\infty} E_{i} . \end{matrix}$

Then we have $μ (E_{i}) \leq \frac{1}{2^{i}} and μ (H_{k}) \leq \sum_{i = k}^{\infty} \frac{1}{2^{i}} = \frac{1}{2^{k - 1}} .$

If we set $P = ⋂_{k = 1}^{\infty} H_{k} = ⋂_{k = 1}^{\infty} ⋃_{i = k}^{\infty} E_{i} = \underset{i ⟶ \infty}{lim sup} E_{i},$ then we have P ⊆ H _k for each k and μ (P) =0.

Next fix any x ∉ P. Then x ∉ H _k for some k, and therefore x ∉ E _i for all i ≥ k. Hence for any l ≥ i ≥ k we have $\begin{matrix} D ({\bar{g}}_{i} (x), {\bar{g}}_{l} (x)) \\ \leq \sum_{j = i}^{l - 1} D ({\bar{g}}_{j + 1} (x), {\bar{g}}_{j} (x)) \leq \sum_{j = i}^{l - 1} \frac{1}{2^{j}} \leq \frac{1}{2^{i - 1}} . \end{matrix}$

As a consequence, the sequence of fuzzy valued functions $({\bar{g}}_{i} (x))$ is Cauchy. Since $F (R)$ is complete, so it is convergent to a fuzzy valued function. If we define $\bar{f} (x) = {\begin{matrix} I - lim_{n ⟶ \infty} {\bar{g}}_{i} (x) & if the limit exists, \\ \bar{0} & otherwise . \end{matrix}$

Then $\bar{f}$ is measurable and since the limit exists for each x ∉ P we have that ${\bar{f}}_{n} I ⟶ \bar{f}$ pointwise almost everywhere.

Now we show that $({\bar{g}}_{i})$ I-converges in measure to $\bar{f} .$ Fix any i. If x ∉ H _i, then we have $D ({\bar{g}}_{i} (x), {\bar{g}}_{l} (x)) \leq \frac{1}{2^{i - 1}}$ for all l ≥ i. Hence $\begin{matrix} D ({\bar{g}}_{i} (x), \bar{f} (x)) \\ \leq D ({\bar{g}}_{i} (x), I - lim_{l \to \infty} {\bar{g}}_{l} (x)) \\ = I - lim_{l \to \infty} (D ({\bar{g}}_{i} (x), {\bar{g}}_{l} (x)) \leq \frac{1}{2^{i - 1}} . \end{matrix}$

Therefore ${i \in ℕ : D ({\bar{g}}_{i}, \bar{f}) \geq \frac{1}{2^{i - 1}}} \subseteq H_{k},$ so we have $\begin{matrix} μ (D ({\bar{g}}_{i}, \bar{f}) \geq \frac{1}{2^{i - 1}}) \\ \leq μ (H_{i}) \leq \frac{1}{2^{i - 1}} ⟶ 0 as i ⟶ \infty \end{matrix}$

It follows from this that ${\bar{g}}_{i} mI ⟶ \bar{f} .$ Thus we have shown that $({\bar{g}}_{l})$ is a subsequence of $({\bar{f}}_{n})$ that I-converges in measure to $\bar{f} .$ Combine this with the fact that $({\bar{f}}_{n})$ is ideally Cauchy in measure to show that ${\bar{f}}_{n} mI ⟶ \bar{f} .$

(b) ⇒ (a) It is easy, so omitted.

6 Conclusions

Fuzzy numbers is an open source package for software environment for statistical computing and graphics, which includes an implementation of a very powerful and quite popular high-level language. Based on these studies, we can further probe the applications of fuzzy soft sets in different fields such as image processing, pattern recognition, medical diagnosis, region extraction, data analysis, decision making and coding theory. Ansastassiou and Duman [2] proved Korovkin’s approximation theorem for fuzzy positive linear operators and estimated the rates of statistical fuzzy convergence based on a non-negative regularly summable matrix of operators via the fuzzy modulus of continuous of fuzzy number valued functions. Further applications of fuzzy numbers we refer [12 , 20–22] and applications of soft sets reader can see [5 , 30–32].

Footnotes

Acknowledgments

The author thank associate editor and the referees for their valuable comments and helpful suggestions for improvement of the article.

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