Egorov’s Theorem Via Deferred Cesàro Statistical Convergence in Measure for Fuzzy Valued Sequences

Abstract

In this article, we introduce and study deferred Cesàro statistical convergence in measure by virtue of sequences of fuzzy valued measurable functions. We define inner and outer deferred Cesàro statistical convergence in measure for sequences of fuzzy valued measurable functions and exhibit that in a finite measurable set, both types of convergence are equal. Finally, we prove the new version of Egorov’s theorem by using deferred Cesàro statistical convergence for the sequences of fuzzy valued functions in finite measurable spaces.

Keywords

Egorov’s theorem fuzzy valued function measure space statistical convergence

1. Introduction and Preliminaries

The notion of fuzzy sets and its arithmetic operations was initially presented by Zadeh (1965) in 1965. After that, many mathematician studied the applications of fuzzy sets by introducing fuzzy topology, fuzzy ordering, etc. In addition, Matloka (1986) studied the theory of a sequences of fuzzy numbers. Many authors such as Tripathy and Nanda (2000), Savaş and Mursaleen (2004) and Hazarika and Savaş (2011) discussed the theory of sequences of fuzzy numbers.

A fuzzy number $\bar{y} : R \to [0, 1]$ is a fuzzy set of real numbers which fulfills the given assertions:

(1)
normal, that is $\bar{y} (υ) = 1$ for some $υ \in R,$
(2)
fuzzy convex, that is $\bar{y} (υ) \geq min {\bar{y} (m), \bar{y} (t)}$ where $m < υ < t$ ,
(3)
$\bar{y}$ is upper semi continuous,
(4)
the closure of ${\bar{y}}^{0} = {υ \in R : \bar{y} (υ) > 0}$ is compact.
Also, the $α -$ level cut of a fuzzy number $\bar{y}$ for $α \in (0, 1],$ can be defined as
$[\bar{y}]^{α} = {υ \in R : \bar{y} (υ) \geq α} .$
Hence, $\bar{y}$ is a fuzzy number iff $[\bar{y}]^{α}$ is a closed interval and $[\bar{y}]^{1} \neq ϕ .$ Throughout the paper, $F (R)$ is the space of all fuzzy numbers.

Let ${\bar{y}}_{1}$ and ${\bar{y}}_{2}$ be two fuzzy numbers. Then the distance between ${\bar{y}}_{1}$ and ${\bar{y}}_{2}$ which is defined by Matloka in Matloka (1986) is given as
$\begin{aligned} d ({\bar{y}}_{1}, {\bar{y}}_{2}) & = sup_{0 \leq α \leq 1} d_{H} ({\bar{y}}_{1}, {\bar{y}}_{2}), \end{aligned}$
where $d : F (R) \times F (R) \to [0, \infty)$ and $d_{H}$ is Hausdorff metric defined as

$\begin{aligned} d_{H} ({\bar{y}}_{1}, {\bar{y}}_{2}) & = max {| ({\bar{y}}_{1}^{α})^{-} - ({\bar{y}}_{2}^{α})^{-} |, | ({\bar{y}}_{1}^{α})^{+} - ({\bar{y}}_{2}^{α})^{+} |} . \end{aligned}$
$({\bar{y}}_{1}^{α})^{-}$ and $({\bar{y}}_{2}^{α})^{+}$ are the lower and upper bound of $α -$ level cut. For any ${\bar{y}}_{1}, {\bar{y}}_{2}, {\bar{y}}_{3}, {\bar{y}}_{4} \in F (R),$ we have (1)
$(F (R), d)$ is a complete metric space,
(2)
$d (γ {\bar{y}}_{1}, γ {\bar{y}}_{2}) = | γ | d ({\bar{y}}_{1}, {\bar{y}}_{2}), γ \in R,$
(3)
$d ({\bar{y}}_{1} + {\bar{y}}_{4}, {\bar{y}}_{2} + {\bar{y}}_{4}) = d ({\bar{y}}_{1}, {\bar{y}}_{2}),$
(4)
$d ({\bar{y}}_{1} + {\bar{y}}_{3}, {\bar{y}}_{2} + {\bar{y}}_{4}) \leq d ({\bar{y}}_{1}, {\bar{y}}_{2}) + d ({\bar{y}}_{3}, {\bar{y}}_{4}) .$
For detailed study on fuzzy sequence spaces one may refer (Jasrotia et al., 2020; Mohiuddine & Lohani, 2009; Mursaleen & Mohiuddine, 2009; Nanda, 1989; Raj et al., 2012; Raj & Pandoh, 2019).
Lemma 1.1
Negoita and Ralescu (1989) Let $\bar{y} \in F (R)$ and $[\bar{y}]^{α} = [({\bar{y}}^{α})^{-}, ({\bar{y}}^{α})^{+}],$ then

(1)
$({\bar{y}}^{α})^{-}$ and $({\bar{y}}^{α})^{+}$ are right continuous at $α = 0,$
(2)
$({\bar{y}}^{α})^{-}$ and $({\bar{y}}^{α})^{+}$ are left continuous non-decreasing and right non-decreasing functions, respectively on $(0, 1],$
(3)
$({\bar{y}}^{1})^{-} \leq ({\bar{y}}^{1})^{+} .$

The concept of statistical convergence was initially presented by Fast (1951) and Schoenberg (1959) independently. Later on, Rath and Tripathy (1996) studied statistical convergence from sequence spaces point of view. In 1980 Gadjiev and Orhan (2002) studied the order of statistical convergence for a sequence of numbers.

As defined by Nuray and Savaş (1995), a sequence $\bar{y} = ({\bar{y}}_{k})$ of fuzzy numbers is said to be statistical convergent to a fuzzy number ${\bar{y}}_{0}$ , if
$lim_{t \to \infty} \frac{1}{t} | {k \leq t : d ({\bar{y}}_{k}, {\bar{y}}_{0}) \geq ε} | = 0$
for each $ε > 0,$ where the vertical bar implies cardinality of the enclosed set. Recently, Hazarika et al. define statistical convergence for double sequences of fuzzy valued measurable functions in Hazarika et al. (2020). Recently, the statistical convergence in measure is studied by Savaş (see Savaş, 2020, Savaş, 2020, Savaş, 2022). For more results on statistical convergence one may refer (Esi & Savas, 2015, Et et al., 2019, Mohiuddine et al., 2012).

Suppose that $l = (l_{x})$ and $m = (m_{x})$ are the sequences of non-negative integers fulfilling

(1)
$l_{x} < m_{x}, \forall x \in N,$
(2)
$lim_{x \to \infty} m_{x} = \infty .$
Then the deferred Cesàro mean of real-valued sequence $y = (y_{k})$ as defined by R. P. Agnew in Agnew (1932) is defined as

$\begin{aligned} (D_{l, m} y)_{x} & = \frac{1}{(m_{x} - l_{x})} \sum_{k = l_{x} + 1}^{m_{x}} y_{k} . \end{aligned}$
Let $A \subset N$ and $x \in N .$ Then, we say that $D δ_{l, m}^{x} (A)$ is $x^{t h}$ partial deferred Cesàro density of $A$ and is defined as
$\begin{aligned} D δ_{l, m}^{x} (A) & = \frac{| A \cap (l_{x}, m_{x}] |}{(m_{x} - l_{x})} . \end{aligned}$
Also, $D δ_{l, m} (A)$ is said to be a deferred Cesàro density or $D_{l, m} -$ density of $A$ where
$\begin{aligned} D δ_{l, m} (A) & = lim_{x \to \infty} \frac{1}{(m_{x} - l_{x})} | {l_{x} < k \leq m_{x} : k \in A} | that is, D δ_{l, m} (A) = lim_{x \to \infty} D δ_{l, m}^{x} (A) \end{aligned}$
exists. The zero density set is given as
$\begin{aligned} Λ & = {A \subset N : D δ_{l, m} (A) = 0} . \end{aligned}$
A sequence $\bar{y} = ({\bar{y}}_{t})$ of fuzzy numbers is said to be deferred Cesàro statistically convergent to a fuzzy number ${\bar{y}}_{0}$ if, $\forall ε > 0$

$\begin{aligned} D δ_{l, m} ({t \in N : d ({\bar{y}}_{t} - {\bar{y}}_{0}) \geq ε}) & = 0 \end{aligned}$
and it is denoted as
$\begin{aligned} D S_{l, m} - lim_{t \to \infty} {\bar{y}}_{t} & = {\bar{y}}_{0} or {\bar{y}}_{t} \overset{D S_{l, m}}{\to} {\bar{y}}_{0} . \end{aligned}$
Throughout the paper, we write almost everywhere as a.e. and fuzzy valued function as FVF, respectively.
2. Main Result

Definition 2.1

The sequence $({\bar{p}}_{t})$ of FVF is pointwise deferred Cesàro statistically convergent to $\bar{p}$ FVF on $[b, c]$ if for every $y \in [b, c]$ and $ε > 0, \exists T_{y} \in Λ$ such that $\forall t \in N ∖ T_{y},$ we have $d ({\bar{p}}_{t} (y), \bar{p} (y)) < ε,$ where $\bar{p}$ is said to be the deferred Cesàro pointwise statistically limit function of $({\bar{p}}_{t}) .$ We may also write deferred Cesàro pointwise statistically convergent as ${\bar{p}}_{t} \overset{p D S_{l, m}}{\to} \bar{p},$ or $p D S_{l, m} - lim {\bar{p}}_{t} (y) = \bar{p} (y) .$ Thus, we say ${\bar{p}}_{t} \overset{p D S_{l, m}}{\to} \bar{p}$ if for every $ε > 0,$

D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ε}) = 0

\forall y \in [b, c] .

Example 2.2

Let $[0, 1]$ be the interval under consideration, and define a sequence ${p_{t}}$ of fuzzy-valued functions on $[0, 1]$ as follows:

p_{t} (y) = {\begin{cases} y, & if t is even, \\ 0, & if t is odd, \end{cases} for all y \in [0, 1] .

Define the function

p (y) = y

for all

y \in [0, 1]

. Let the deferred Cesàro statistical parameters be given by

l_{t} = 2 t

and

m_{t} = t

. We claim that

{p_{t}}

is pointwise deferred Cesàro statistically convergent to

p

[0, 1]

, that is,

p_{t} \overset{p D S l, m}{\to} p on [0, 1] .

Definition 2.3

The sequence $({\bar{p}}_{t})$ of FVF is uniformly deferred Cesàro statistical convergent to $\bar{p}$ FVF on $[b, c]$ if $\forall ε > 0, \exists T \in Λ$ such that $\forall t \in N ∖ T,$ we get $d ({\bar{p}}_{t} (y), \bar{p} (y)) < ε, \forall y \in [b, c] .$ We may also write uniformly deferred Cesàro statistical convergent as ${\bar{p}}_{t} \overset{u D S_{l, m}}{⇉} \bar{p},$ or $u D S_{l, m} - lim {\bar{p}}_{t} (y) = \bar{p} (y) .$ Thus, we say ${\bar{p}}_{t} \overset{u D S_{l, m}}{⇉} \bar{p}$ if for all $ε > 0,$

D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ε}) = 0

\forall y \in [b, c] .

Example 2.4

Let $[0, 1]$ be the interval under consideration, and define a sequence ${p_{t}}$ of real-valued functions on $[0, 1]$ by

p_{t} (y) = {\begin{cases} y^{2}, & if t is even, \\ 0, & if t is odd, \end{cases} for all y \in [0, 1] .

Let

p (y) = y^{2}

for all

y \in [0, 1]

. Define the deferred Cesàro statistical parameters as

l_{t} = 2 t

and

m_{t} = t

. We assert that

{p_{t}}

is uniformly deferred Cesàro statistically convergent to

p

[0, 1]

, that is,

Definition 2.5

The sequence $({\bar{p}}_{t})$ of FVF is deferred Cesàro equi-statistically convergent to $\bar{p}$ FVF if for all $y \in [b, c]$ and $ε > 0, \exists G_{x, ε} = D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ε})$ is uniformly convergent to zero function. We may also write deferred Cesàro equi-statistically convergent as ${\bar{p}}_{t}$ $\bar{p} .$ Thus, we say ${\bar{p}}_{t}$ $\bar{p}$ iff for every $ε, α > 0, \exists k \in N, \forall x \geq k$ and $y \in [b, c]$

D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ε}) < α .

Suppose that $(Ω, M, ζ)$ is a finite measure space. For $A \subseteq M$ we write ${\bar{p}}_{t | A}$ as a sequence of FVF except on some subset $A$ of arbitrarily small measure and ${\bar{p}}_{| A}$ as a FVF except on some subset $A$ of arbitrarily small measure, respectively.

Example 2.6

Let $(Ω, M, ζ)$ be a finite measure space with $Ω = [0, 1]$ and $ζ$ the Lebesgue measure. Define a sequence of fuzzy-valued functions ${p_{t}}$ on $Ω$ by

p_{t} (y) = {\begin{cases} [y - \frac{1}{t} - r, y - \frac{1}{t}, y - \frac{1}{t} + r], & y \in [0, 1 - \frac{1}{t}], \\ [- r, 0, r], & y \in (1 - \frac{1}{t}, 1], \end{cases}

where

r > 0

is fixed. Let the limit fuzzy-valued function be

p (y) = [y - r, y, y + r], \forall y \in [0, 1] .

The deferred Cesàro parameters as

l_{t} = 2 t

and

m_{t} = t

. Then for every

ε, α > 0

, there exists

k \in N

such that for all

x \geq k

and

y \in [0, 1]

D_{l, m}^{δ} ({t \in N : d (p_{t} (y), p (y)) \geq ε}) < α .

Hence,

p_{t} \overset{e D S l, m}{\to} p

[0, 1]

, i.e.,

{p_{t}}

is deferred Cesàro equi-statistically convergent to

p

Remark 2.7

We can say that ${\bar{p}}_{t} \overset{p D S_{l, m}}{\to} \bar{p}$ iff $\forall y \in Ω$ and $\forall$ $ε, α > 0, \exists x \in N,$

D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ε}) < α .

In such case, we put

α = ε .

Thus,

{\bar{p}}_{t}

\bar{p}

implies that

{\bar{p}}_{t} \overset{p D S_{l, m}}{\to} \bar{p} .

Moreover,

{\bar{p}}_{t} \overset{u D S_{l, m}}{⇉} \bar{p}

implies that

{\bar{p}}_{t}

\bar{p} .

Theorem 2.8

Suppose that $(Ω, M, ζ)$ is a finite measure space. Consider $\bar{p}$ is a measurable FVF and $({\bar{p}}_{t})$ is sequence of measurable FVF defined a.e. on $Ω .$ If ${\bar{p}}_{t} \overset{p D S_{l, m}}{\to} \bar{p}$ a.e. on $Ω,$ then $\exists A \subseteq M$ such that $ζ (Ω ∖ A) < ε, \forall ε > 0$ and ${\bar{p}}_{t | A}$ ${\bar{p}}_{| A}$ on $A .$

Proof.

Let us consider that both $({\bar{p}}_{t})$ and $\bar{p}$ are defined a.e. on $Ω$ and ${\bar{p}}_{t} (z) \overset{p D S_{l, m}}{\to} \bar{p} (y), \forall y \in Ω .$ Now, choose $ν, x \in N,$ we see that

\begin{aligned} I & = {y \in Ω : D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq \frac{1}{ν}}) < \frac{1}{ν}} \end{aligned}

is measurable.

The function $Ψ_{t} (y) = d ({\bar{p}}_{t} (y), \bar{p} (y))$ is measurable. Let $P_{t} = Ψ_{t}^{- 1} ([\frac{1}{ν}, \infty)) .$ For all $y \in Ω,$ we have $y \in I$ iff

\frac{1}{(m_{x} - l_{x})} \sum_{n = l_{x} + 1}^{m_{x}} χ P_{t} (y) < \frac{1}{ν} .

Let the function

\begin{aligned} p & = \frac{1}{(m_{x} - l_{x})} \sum_{t = l_{x} + 1}^{m_{x}} χ P_{t} (y) \end{aligned}

is measurable, so we get

I = p^{- 1} ((- \infty, \frac{1}{ν})) .

For

k \in N,

we have

\begin{aligned} Θ_{ν, k} & = {y \in Ω : \forall x \geq k D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq \frac{1}{ν}}) < \frac{1}{ν}} . \end{aligned}

Then, from above observation we conclude that

Θ_{ν, k}

is measurable and thus we have

Θ_{ν, k} \subset Θ_{ν, k + 1} (\forall k \in N) and Ω = ⋃_{k = 1}^{\infty} Θ_{ν, k} .

Consequently,

ζ (Ω) = lim_{t \to \infty} ζ (Θ_{ν, k}) .

Let

ε > 0

be given. For

k \in N,

choose

k (ν) \in N,

such that

ζ (Ω ∖ Θ_{ν, k (ν)}) < \frac{ε}{2^{ν}} .

Fix

A_{0}

such that

A_{0} = ⋃_{ν = 1}^{\infty} (Ω ∖ Θ_{ν, k (ν)}) .

We have

\begin{aligned} ζ (A_{0}) & \leq \sum_{ν = 1}^{\infty} ζ (Ω ∖ Θ_{ν, k (ν)}) < ε . \end{aligned}

Let

A = Ω ∖ A_{0} = ⋂_{ν = 1}^{\infty} Θ_{ν, k (ν)} .

Thus,

ζ (Ω ∖ A) = ζ (A_{0}) < ε .

Therefore,

\forall ν \in N x \geq k (ν)

and

y \in A,

we have

\begin{aligned} D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) > \frac{1}{ν}}) & < \frac{1}{ν} . \end{aligned}

This proves that

{\bar{p}}_{t | A}

{\bar{p}}_{| A}

A .

Corollary 2.9

Suppose that $(Ω, M, ζ)$ is a finite measure space. Consider $\bar{p}$ is a measurable FVF and $({\bar{p}}_{t})$ is sequence of measurable FVF defined a.e. on $Ω,$ then ${\bar{p}}_{t} \overset{p D S_{l, m}}{\to} \bar{p}$ a.e. on $Ω$ iff there exists a sequence $(A_{t})$ of sets on $M$ such that ${\bar{p}}_{t | A_{t}}$ ${\bar{p}}_{| A_{t}}$ on $A_{t}, \forall t$ and $ζ (Ω ∖ \cup_{t \in N} A_{t}) = 0.$

Proof.

Let us consider that both FVF $\bar{p}$ and sequence of FVF $({\bar{p}}_{t})$ are measurable and are defined a.e. on $Ω$ and ${\bar{p}}_{t} \overset{p D S_{l, m}}{\to} \bar{p}$ a.e. on $Ω .$ Then, by considering $ε = \frac{1}{t}$ in theorem $2.5$ we get the conclusion. Now, let ${\bar{p}}_{t | A_{t}}$ ${\bar{p}}_{| A_{t}}$ on $A_{t} .$ Then we get ${\bar{p}}_{t | A_{t}} \overset{p D S_{l, m}}{\to} {\bar{p}}_{| A_{t}}$ on $A_{t} .$ Hence, we get ${\bar{p}}_{t} \overset{p D S_{l, m}}{\to} \bar{p}$ a.e. on $Ω .$

Definition 2.10

Let $(Ω, M, ζ)$ be measurable space. Also, consider set $T^{0}$ of all Fuzzy valued measurable function (FVMF) defined a.e. on $Ω$ such that $({\bar{p}}_{t})$ and $\bar{p}$ are in $T^{0} .$ Then, the outer deferred Cesàro Statistical convergence in measure of a sequence $({\bar{p}}_{t})$ of FVMF to $\bar{p}$ FVMF is defined by

D δ_{l, m}^{x} ({t \in N : ζ ({y \in Ω : d ({\bar{p}}_{t} (y), p (y)) > ν}) \geq η}) \to 0, if x \to \infty

(2.1)

\forall ν, η > 0.

In this case, we write

{\bar{p}}_{t} \overset{D δ_{l, m}, ζ}{\to} \bar{p} .

Example 2.11

Let the measurable space $(Ω, M, ζ)$ be defined with $Ω = [0, 1]$ and $ζ$ as the Lebesgue measure. Define a sequence of fuzzy-valued measurable functions ${p_{t}}$ on $Ω$

p_{t} (y) = [y - \frac{1}{t}, y, y + \frac{1}{t}], for all y \in [0, 1] .

The limiting function is

p (y) = [y, y, y], for all y \in [0, 1],

which is a crisp function since the endpoints coincide. Now, for small

ν > 0

, consider the set

A_{t} (ν) = {y \in Ω : d (p_{t} (y), p (y)) > ν},

where

d

is a suitable metric between fuzzy numbers. As

t

increases, the width of

p_{t} (y)

decreases, and

d (p_{t} (y), p (y)) \to 0

. Therefore, for all large

t

, we have

ζ (A_{t} (ν)) < η

for any

η > 0

. Thus, the set

{t \in N : ζ ({y \in Ω : d (p_{t} (y), p (y)) > ν}) \geq η}

has deferred Cesàro density tending to

0

x \to \infty

. That is,

D_{l, m}^{δ} ({t \in N : ζ ({y \in Ω : d (p_{t} (y), p (y)) > ν}) \geq η}) \to 0.

Hence, the sequence

{p_{t}}

is outer deferred Cesàro statistically convergent in measure to

p

, and we write:

p_{t} \overset{D_{l, m, ζ}^{δ}}{\to} p .

Remark 2.12

By replacing the order of $D δ_{l, m}^{x}$ and $ζ$ in $(2.1),$ we get the inner deferred Cesáro statistical convergence in measure of a sequence $({\bar{p}}_{t})$ of FVMF to $\bar{p}$ FVMF as

ζ ({y \in Ω : D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), p (y)) \geq ν}) \geq η)} \to 0, if x \to \infty .

In this case, we say that

{\bar{p}}_{t} \overset{ζ, D δ_{l, m}}{\to} \bar{p} .

Theorem 2.13

Suppose that $(Ω, M, ζ)$ be a measure space and $\bar{p}, ({\bar{p}}_{t}) \in T^{0} .$ Then (1)

If ${\bar{p}}_{t} \overset{D δ_{l, m}, ζ}{\to} \bar{p},$ then ${\bar{p}}_{t} \overset{ζ, D δ_{l, m}}{\to} \bar{p} .$

(2)

If ${\bar{p}}_{t} \overset{ζ, D δ_{l, m}}{\to} \bar{p},$ then ${\bar{p}}_{t} \overset{D δ_{l, m}, ζ}{\to} \bar{p}$ provided that $ζ (Ω) < \infty .$

Proof.

(i) We know that $D δ_{l, m}^{x} : G_{1} \to [0, 1]$ is a probability measure where $G_{1} \subseteq N .$ Let us consider the product measure $ζ \times D δ_{l, m}^{x}$ on product algebra $M \times G_{1}$ of subsets $Ω \times N .$ Fix $ν > 0,$ we have

U_{ν} = {(y, t) \in Ω \times N : d ({\bar{p}}_{t} (y), p (y)) \geq ν} .

We define

ϕ : Ω \times N \to R

ϕ (y, t) = d ({\bar{p}}_{t} (y), p (y))

M \times G_{1} -

measurable. Therefore,

U_{ν} \in M \otimes G_{1} .

Now, for any

B \subset Ω \times N

we have

B (y) = {t \in N : (y, t) \in B} if y \in Ω and B (t) = {t \in N : (y, t) \in B} .

Firstly, we show that

\forall ε, ν > 0, \exists x_{0} \in N

and

\forall x \geq x_{0}

ζ ({y \in Ω : D δ_{l, m}^{x} (U_{ν} (y) \geq ν)}) < ε .

(2.2)

Set

ε, η > 0.

Since

{\bar{p}}_{t} \overset{D δ_{l, m}, ζ}{\to} \bar{p},

one may have

x_{0} \in N

such that

x \geq x_{0},

we get the following

D δ_{l, m}^{x} ({t \in N : ζ (U_{ν} (t)) \geq 1}) < \frac{η}{2}

(2.3)

and

D δ_{l, m}^{x} ({t \in N : ζ (U_{ν} (t)) \geq \frac{η ε}{4}}) < \frac{η ε}{4} .

(2.4)

Let

E = {t \in N : ζ (U_{ν} (t)) < 1} .

Then, we have from (2.3) that

D δ_{l, m}^{x} (N ∖ E) < \frac{η}{2},

$\forall x \geq x_{0} .$ Hence, $\forall x \geq x_{0}$ we get

\begin{aligned} ζ ({y \in Ω : D δ_{l, m}^{x} (U_{ν} (y)) \geq η)}) & \leq ζ ({y \in Ω : D δ_{l, m}^{x} (U_{ν} (y) \cap E) \geq \frac{η}{2}}) \\ + ζ ({y \in Ω : D δ_{l, m}^{x} (U_{ν} (y) ∖ E) \geq \frac{η}{2}}) \\ \leq ζ ({y \in Ω : D δ_{l, m}^{x} (U_{ν} (y) \cap E) \geq \frac{η}{2}}) . \end{aligned}

Suppose that

U_{ν}^{*} = U_{ν} \cap (Ω \times E) .

Therefore, we get

U_{ν}^{*} (y) = U_{ν} (y) \cap E, y \in Ω and U_{ν}^{*} (t) = U_{ν} (t), t \in E .

To find the relation (2.2), we show that

ζ ({y \in Ω : D δ_{l, m}^{x} (U_{ν}^{*} (y)) \geq \frac{η}{2}}) < ε, \forall x \geq x_{0} .

(2.5)

For

U_{ν}^{*} \subset Ω \times E

and for fix

x \in N,

Fubini’s theorem applies to the characteristic function of

U_{ν}^{*}

if it has finite measure.

ζ \times D δ_{l, m}^{x} .

We have

U_{ν}^{*} = ⋃_{t \in E} (t \times U_{ν} (t)),

where

ζ (U_{ν} (t)) < 1, \forall t \in E

and

D δ_{l, m}^{x} ({t}) = 0, \forall t > x .

Thus,

\begin{aligned} \int_{E} ζ (U_{ν}^{*} (t)) d t & = (ζ \times D δ_{l, m}^{x}) (U_{η}^{*}) = \int_{Ω} D δ_{l, m}^{x} (U_{η}^{*} (y)) d y . \end{aligned}

Suppose that

x_{0} \in N

such that

x \geq x_{0},

we have

\begin{aligned} \frac{η ε}{2} & > \frac{η ε}{4} + D δ_{l, m}^{x} ({t \in N : ζ (U_{ν} (t)) \geq \frac{η ε}{4}}) \\ \geq \int_{{t \in E : ζ (U_{ν} (t)) < \frac{η ε}{4}}} ζ (U_{ν} (t)) d t + \int_{{t \in E : ζ (U_{η} (t)) \geq \frac{η ε}{4}}} 1 d t \\ \geq \int_{E} ζ (U_{ν} (t)) d t \\ = \int_{E} ζ (U_{ν}^{*} (t)) d t \\ = \int_{Ω} D δ_{l, m}^{x} (U_{ν}^{*} (y)) d y \\ \geq \int_{{y \in Ω : ζ (U_{ν}^{*} (y)) \geq \frac{η}{2}}} D δ_{l, m}^{x} (U_{ν}^{*} (y)) d y \\ \geq \frac{η}{2} ζ ({y \in Ω : D δ_{l, m}^{x} (U_{ν}^{*} (y)) \geq \frac{η}{2}}), \end{aligned}

which proves that (2.5) is true.

(ii) Let $ζ (Ω) < \infty .$ Fix $ν > 0.$ We need to show that $\forall ε, η > 0, \exists x_{0} \in N$ and $\forall x \geq x_{0}$

D δ_{l, m}^{x} ({t \in N : ζ (U_{ν} (t)) \geq η}) < ε .

Since

ε, η > 0

and

{\bar{p}}_{t} \overset{ζ, D δ_{l, m}}{\to} \bar{p},

we have

x_{0} \in N

such that

\forall x \geq x_{0},

we have

\begin{aligned} ζ ({y \in Ω : D δ_{l, m}^{x} (U_{ν} (y)) \geq \frac{η ε}{2 ζ (Ω)}}) < \frac{η ε}{2} . \end{aligned}

By applying Fubini theorem for the characteristic function of

U_{ν} \subset Ω \times N,

we have

\begin{aligned} \int_{Ω} D δ_{l, m}^{x} (U_{ν} (y)) d y & = (ζ \times D δ_{l, m}^{x}) (U_{ν}) \\ = \int_{N} ζ (U_{ν} (t)) d t . \end{aligned}

Suppose

x_{0} \in N

such that

\forall x > x_{0},

we have

\begin{aligned} η ε & > \frac{η ε μ (Ω)}{2 ζ (Ω)} + ζ ({y \in Ω : D δ_{l, m}^{x} (U_{ν} (y)) \geq \frac{η ε}{2 ζ (Ω)}}) \\ \geq \int_{{y \in Ω : D δ_{l, m}^{x} (U_{ν} (y)) \leq \frac{η ε}{2 ζ (Ω)}}} D δ_{l, m}^{x} (U_{ν}^{x} (y)) d y + \int_{{y \in Ω : D δ_{l, m}^{x} (U_{ν} (y)) \geq \frac{η ε}{2 ζ (Ω)}}} 1 d y \\ \geq \int_{Ω} D δ_{l, m}^{x} (U_{ν} (y)) d y \\ = \int_{N} ζ (U_{ν} (t)) d t \\ \geq \int_{{t \in N : ζ (U_{ν} (t)) \geq η}} ζ (U_{ν} (t)) d t . \\ \geq η D δ_{l, m}^{x} ({t \in N : ζ (U_{ν} (t)) \geq η}) . \end{aligned}

Example 2.14

Let $(Ω, M, ζ)$ be a measure space where $Ω = [0, \infty)$ and $ζ$ is the Lebesgue measure. Thus, $ζ (Ω) = \infty$ . Define a sequence ${p_{t}}$ of fuzzy-valued measurable functions on $Ω$ as:

p_{t} (y) = [0, \frac{1}{t}, \frac{2}{t}], \forall y \in Ω,

and define the limiting fuzzy-valued function as:

p (y) = [0, 0, 0], \forall y \in Ω .

Here, each

p_{t} (y)

is a triangular fuzzy number that becomes narrower as

t \to \infty

and converges to the crisp fuzzy number

p (y)

. Fix any

ν > 0

. For sufficiently large

t

, the distance

d (p_{t} (y), p (y)) < ν

for all

y \in Ω

. Hence, for any

ϵ > 0

, the set

{t \in N : ζ ({y \in Ω : d (p_{t} (y), p (y)) \geq ν}) > ϵ}

becomes small in the deferred Cesàro statistical sense with respect to sequences

{l_{t}}

and

{m_{t}}

. Thus,

p_{t} \overset{ζ, D_{δ_{l, m}}}{\to} p

. However, since

ζ (Ω) = \infty

, the converse implication, that is,

p_{t} \overset{D_{δ_{l, m}}, ζ}{\to} p,

does not necessarily hold. The integration over an infinite measure space does not guarantee convergence in the deferred Cesàro sense. Therefore, this example shows that the converse of Theorem 2.12 fails when

ζ (Ω) = \infty

Definition 2.15

Suppose that $(Ω, M, ζ)$ is a finite measure space. Consider $({\bar{p}}_{t})$ and $\bar{p}$ are in $T^{0} .$ A sequence of FVMF $({\bar{p}}_{t})$ is deferred Cesàro statistical convergent in measure (or DCSCM) to a FVMF $\bar{p},$ in symbol ${\bar{p}}_{t} \overset{ζ D S_{l, m}}{\to} \bar{p},$ if the sequence $ζ ({y \in Ω : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν})$ is deferred Cesàro statistical convergent to 0 for $ν > 0.$ This concept is equivalent to the following formula

{t \in N : ζ ({z \in Ω : d ({\bar{p}}_{t} (y), p (y)) \geq ν}) > η} \in Λ, \forall η, ν > 0.

Thus,

η = ν

ν = \frac{1}{x}, x \in N .

Example 2.16

Let $(Ω, M, ζ)$ be a finite measure space with $Ω = [0, 1]$ and $ζ$ the Lebesgue measure. Define a sequence of fuzzy-valued measurable functions ${p_{t}}$ on $Ω$ by:

p_{t} (y) = [y - \frac{1}{t}, y, y + \frac{1}{t}], for all y \in [0, 1] .

Each

p_{t} (y)

represents a triangular fuzzy number whose support width decreases as

t

increases. The limiting function is defined as follows:

p (y) = [y, y, y], for all y \in [0, 1],

which is a crisp fuzzy-valued function. For any

ν > 0

, consider the set:

E_{t} (ν) = {y \in [0, 1] : d (p_{t} (y), p (y)) \geq ν},

where

d

is a suitable metric on fuzzy numbers. As

t

increases,

d (p_{t} (y), p (y)) < ν

for all

y \in [0, 1]

, so

ζ (E_{t} (ν)) \to 0

. Hence, for every

ε > 0

, the set

{t \in N : ζ (E_{t} (ν)) > ε}

belongs to the

Λ

, under suitable choices such as

l_{t} = 2 t

m_{t} = t

. Therefore,

{p_{t}}

is deferred Cesàro statistically convergent in measure to

p

, denoted by:

p_{t} \to_{ζ}^{D S l, m} p .

Proposition 2.17

Suppose that $(Ω, M, ζ)$ is a finite measure space and $({\bar{p}}_{t}), \bar{p} \in T^{0} .$ Then ${\bar{p}}_{t} \overset{u D S_{l, m}}{⇉} \bar{p} \Rightarrow {\bar{p}}_{t} \overset{ζ D S_{l, m}}{\to} \bar{p} .$

Proof.

Let ${\bar{p}}_{t} \overset{u D S_{l, m}}{⇉} \bar{p} .$ Let $ν > 0,$ then there exists a set $L \in Λ$ such that $d ({\bar{p}}_{t} (y), \bar{p} (y)) < ν, \forall t \notin L, y \in Ω .$

Thus, we get

\begin{aligned} {t \in N : ζ ({y \in Ω : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν}) \geq ν} & \subset {t \in N : ζ ({y \in Ω : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν}) \neq ϕ} \\ \subset L \in Λ . \end{aligned}

This proves that

{\bar{p}}_{t} \overset{ζ D S_{l, m}}{\to} \bar{p} .

Theorem 2.18

Suppose that $(Ω, M, ζ)$ is a measure space and $({\bar{p}}_{t}), \bar{p} \in T^{0} .$ If sequence $({\bar{p}}_{t})$ of FVMF is pointwise deferred Cesàro statistically convergent to $\bar{p}$ FVMF a.e. on $Ω,$ then ${\bar{p}}_{t} \overset{μ D S_{l, m}}{\to} \bar{p} .$

Proof.

Let ${\bar{p}}_{t} (y) \overset{ζ D S_{l, m}}{\to} \bar{p} (y)$ a.e. on $Ω .$ From theorem (2.8) ${\bar{p}}_{t} \overset{ζ, D δ_{l, m}}{\to} \bar{p}$ is same as ${\bar{p}}_{t} (y) \overset{μ D S_{l, m}}{\to} \bar{p} (y) .$ So we first show that ${\bar{p}}_{t} (y) \overset{ζ, D δ_{l, m}}{\to} \bar{p} (y) .$ Let us consider $ε, q > 0.$ From theorem (2.5), $A \in M$ such that ${\bar{p}}_{t | A}$ ${\bar{p}}_{| A}$ and $ζ (Ω ∖ A) < ε .$ Let $k \in N$ such that

D δ_{l, m}^{x} ({n \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν}) < ν

\forall x \geq k and y \in A .

Thus,

\forall x \geq k

we get

\begin{aligned} {y \in Ω : D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν}) \geq ν} & \subset Ω ∖ A . \end{aligned}

Hence,

ζ ({y \in Ω : D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), p (y)) \geq ν}) \geq ν}) < ε

\forall x \geq k

as desired.

Theorem 2.19

Suppose that $(Ω, M, ζ)$ is a finite measure space and $({\bar{p}}_{t}), \bar{p} \in T^{0} .$ If ${\bar{p}}_{t} \overset{p D S_{l, m}}{\to} \bar{p}$ a.e. on $Ω,$ then ${\bar{p}}_{t} (y) \overset{μ D S_{l, m}}{\to} \bar{p} (y) .$

Proof.

Suppose that $ν, ε > 0.$ From theorem (2.5), $\exists A \subset Ω$ such that ${\bar{p}}_{t | A}$ ${\bar{p}}_{| A}$ on $A$ and $ζ (Ω ∖ A) < ε .$ Let $k \in N$ such that

D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν}) \leq ν, (\forall x \geq k and y \in A)

which exhibits that

{y \in Ω : D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν}) \geq ν} \subset Ω ∖ A, \forall (x \geq k) .

Thus,

ζ ({y \in Ω : D δ_{l, m}^{x} ({t \in N : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν}) \geq ν}) < ε .

Corollary 2.20

Suppose that $(Ω, M, ζ)$ is a finite measure space and $({\bar{p}}_{t}), \bar{p} \in T^{0} .$ If ${\bar{p}}_{t} (y) \overset{ζ D S_{l, m}}{\to} \bar{p} (y),$ then $\exists$ a subsequence $({\bar{p}}_{t_{k}})$ and $({\bar{p}}_{t})$ such that ${\bar{p}}_{t_{k}} (y) \overset{p D S_{l, m}}{\to} \bar{p} (y)$ a.e. on $Ω .$

Proof.

Let ${\bar{p}}_{t} (y) \overset{μ D S_{l, m}}{\to} \bar{p} (y),$ so any subsequence $({\bar{p}}_{t_{k}})$ of $({\bar{p}}_{t})$ is also deferred Cesàro statistical convergent in measure to $\bar{p} .$ Also, $({\bar{p}}_{t})$ has a subsequence i.e. deferred Cesàro statistical convergent in measure to $\bar{p}$ a.e. on $Ω .$

Definition 2.21

Suppose that $(Ω, M, ζ)$ is a finite measure space. Consider $({\bar{p}}_{t})_{β}^{+}, ({\bar{p}}_{t})_{β}^{-}, {\bar{p}}_{β}^{+}, {\bar{p}}_{β}^{-} \in T^{0} .$ The sequence $[{\bar{p}}_{t} (y)]_{β}$ is uniformly deferred Cesàro statistically convergent in measure (or UDCSCM) to $[\bar{p} (y)]_{β}$ w.r.t $β$ if the sequence

ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{+} - (\bar{p} (y))_{β}^{+} | \geq ν})

and

ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{-} - (\bar{p} (y))_{β}^{-} | \geq ν})

both are deferred Cesàro statistical convergent in measure to 0 for all

ν > 0.

The above equations are equivalent to following:

\forall η, ν > 0, {t \in N : ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{+} - (\bar{p} (y))_{β}^{+} | \geq ν}) \geq η} \in Λ

as well as

\forall η, ν > 0, {t \in N : ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{-} - (\bar{p} (y))_{β}^{-} | \geq ν}) \geq η} \in Λ .

In such cases, we write

η = ν

ν = \frac{1}{x}, x \in N .

Example 2.22

Let $(Ω, M, ζ)$ be a finite measure space with $Ω = [0, 1]$ and $ζ$ as the Lebesgue measure. Define a sequence ${p_{t}}$ of fuzzy-valued measurable functions on $Ω$

p_{t} (y) = [y - \frac{1 + β}{t}, y, y + \frac{1 + β}{t}], for β \in [0, 1], y \in [0, 1] .

Let the limiting function be

p (y) = [y, y, y], for all y \in [0, 1] .

Then, for each

β \in [0, 1]

(p_{t})_{β}^{+} (y) = y + \frac{1 + β}{t}, (p_{t})_{β}^{-} (y) = y - \frac{1 + β}{t}, p_{β}^{+} (y) = p_{β}^{-} (y) = y .

Thus,

| (p_{t})_{β}^{+} (y) - p_{β}^{+} (y) | = | (p_{t})_{β}^{-} (y) - p_{β}^{-} (y) | = \frac{1 + β}{t} \leq \frac{2}{t} .

Therefore,

sup_{β \in [0, 1]} | (p_{t})_{β}^{+} (y) - p_{β}^{+} (y) | < ν, for large t .

So,

ζ ({y \in Ω : sup_{β \in [0, 1]} | (p_{t})_{β}^{+} (y) - p_{β}^{+} (y) | \geq ν}) \to 0

and similarly for the lower

β

-cut. Hence, the sequence

{p_{t}}

is uniformly deferred Cesàro statistically convergent in measure (UDCSCM) to

p

, and we write:

p_{t} \overset{UDCSCM}{\to} p .

Theorem 2.23

Suppose that $(Ω, M, ζ)$ is a finite measure space and $({\bar{p}}_{t}), \bar{p} \in T^{0} .$ A sequence of FVMF $(\bar{p})_{t}$ is statistically convergent in measure to FVMF $\bar{p}$ iff $[{\bar{p}}_{t} (y)]_{β}$ is uniformly statistically convergent in measure to $[\bar{p} (y)]_{β}$ w.r.t $β .$

Proof.

Let us consider $({\bar{p}}_{t})$ is deferred Cesàro statistically convergent in measure to $\bar{p} .$ Then,

ζ ({y \in Ω : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν})

is deferred Cesàro statistically convergent in measure to zero for every

ν > 0,

i.e.

\begin{aligned} {t \in N : ζ {y \in Ω : d ({\bar{p}}_{t} (y), \bar{p} (y))} \geq ν} \in Λ . \end{aligned}

Thus,

{t \in N : ζ ({y \in Ω : sup_{β \in [0, 1]} max {| ({\bar{p}}_{t} (y))_{β}^{-} - \bar{p} (y)_{β}^{-} |, | ({\bar{p}}_{t} (y))_{β}^{+} - (\bar{p} (y))_{β}^{+} |} \geq ν}) \geq ν} \in Λ .

Therefore,

\forall ν > 0

we get

{t \in N : ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{-} - (\bar{p} (y))_{β}^{-} | \geq ν}) \geq ν} \in Λ

as well as

{t \in N : ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{+} - (\bar{p} (y))_{β}^{+} | \geq ν}) \geq ν} \in Λ,

which obtain

ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{-} - (\bar{p} (y))_{β}^{-} | \geq ν}) \overset{μ D S_{l, m}}{\to} 0

as well as

ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{+} - (\bar{p} (y))_{β}^{+} | \geq ν}) \overset{μ D S_{l, m}}{\to} 0.

Hence,

[{\bar{p}}_{t} (y)]_{β}

is uniform deferred Cesàro statistically convergent in measure to

[\bar{p} (y)]_{β}

w.r.t

β .

Then,

\forall ν > 0

we get

ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{-} - (\bar{p} (y))_{β}^{-} | \geq ν}) \overset{μ D S_{l, m}}{\to} 0

as well as

ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{+} - (\bar{p} (y))_{β}^{+} | \geq ν}) \overset{μ D S_{l, m}}{\to} 0.

Thus,

{t \in N : ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{-} - (\bar{p} (y))_{β}^{-} | \geq ν}) \geq ν} \in Λ

as well as

{t \in N : ζ ({y \in Ω : sup_{β \in [0, 1]} | ({\bar{p}}_{t} (y))_{β}^{+} - (\bar{p} (y))_{β}^{+} | \geq ν}) \geq ν} \in Λ .

From the above equations, we get

{t \in N : ζ ({y \in Ω : sup_{β \in [0, 1]} max {| ({\bar{p}}_{t} (y))_{β}^{-} - \bar{p} (y)_{β}^{-} |, | ({\bar{p}}_{t} (y))_{β}^{+} - (\bar{p} (y))_{β}^{+} |} \geq ν}) \geq ν} \in Λ,

which obtains

{t \in N : ζ ({y \in Ω : d ({\bar{p}}_{t} (y), \bar{p} (y)) \geq ν}) \geq ν} \in Λ .

which implies that

({\bar{p}}_{t})

is deferred Cesàro statistically convergent in measure to FVMF

\bar{p} .

Footnotes

Acknowledgements

The corresponding author thanks the Council of Scientific and Industrial Research (CSIR), India for partial support under Grant No. 25(0288)/18/EMR-II, dated 24/05/2018.

ORCID iDs

Sanjeev Verma

Swati Jasrotia

Kuldip Raj

Mohammad Mursaleen

Author Contributions

All authors jointly worked on deriving the results and writing the manuscript and approved the final version of manuscript.

Ethical Considerations

This article does not contain any studies with human participants or animals performed by any of the authors.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Not applicable.

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