A NEW APPROACH TO EGOROV’S THEOREM BY MEANS OF 𝛼𝛽-STATISTICAL IDEAL CONVERGENCE Текст научной статьи по специальности «Математика»

72

Probl. Anal. Issues Anal. Vol. 12 (30), No 1, 2023, pp. 72-86

DOI: 10.15393/j3.art.2023.11890

UDC 510.22

SONALI SHARMA, KULDIP RAJ

A NEW APPROACH TO EGOROV'S THEOREM BY MEANS OF aft-STATISTICAL IDEAL CONVERGENCE

Abstract. In this work, we introduce the «^-statistical pointwise ideal convergence, «^-statistical uniform ideal convergence, and afi-equi-statistical ideal convergence for sequences of fuzzy-valued functions. With the help of some examples, we present the relationship between these convergence concepts. Moreover, we give the «^-statistical ideal version of Egorov's theorem for the sequences of fuzzy valued measurable functions.

Key words: Egorov's theorem, afi-statistical pointwise ideal convergence, afi-statistical uniform ideal convergence, afi-statistical equi-ideal convergence

2020 Mathematical Subject Classification: 40A05, 40A30,

46S40, 47S40

1. Introduction and Preliminaries. The convergence of sequences plays a crucial role in the functional analysis. In the usual convergence, almost all elements of the sequence have to belong to an arbitrarily small neighborhood of the limit, whereas in the statistical one, this condition is relaxed. Basically, statistical convergence demands the validity of the convergence condition only for the majority of elements. The hypothesis of statistical convergence was introduced by Fast [4] and Steinhaus [18], independently. A lot of development has been done in statistical convergence. Generalized statistical convergence for the double sequences was examined by Mursaleen et al. [16]. The ideal convergence is a generalization of the statistical convergence. Balcerzak et al. [3] presented different types of statistical convergence and ideal convergence for sequences of functions. Lacunary statistical convergence in measure for sequences of fuzzy-valued functions was examined by Kisi and Dundar [9]. Recently, Kisi [10] introduced lacunary statistical convergence in measure for double sequences of fuzzy-valued functions.

© Petrozavodsk State University, 2023

In summability theory, ideal convergence became a very important concept. The perception of I—convergence was studied by Kostyrko et al. [15]. The ideal convergence of continuous functionss was given by Jasinski and Reclaw [8]. The ideal convergence by using fuzzy numbers was investigated by Kumar and Kumar [11]. Later on, lacunary ideal convergence for fuzzy real-valued sequences was studied by Hazarika [6]. These concepts were further generalized by Hazarika [7] by using the ideal convergence. To learn more about ideal convergence,see [12], [14] and [19]. The fuzzy-set theory is a beneficial tool in explaining the situation in which there is a lack of data. Zadeh [20] established the idea of the fuzzy set theory, which was further generalized by Matloka [17]. This study of Zadeh attracted many researchers from different fields of sciences and mathematics. Altin et al. [2] examined pointwise statistical convergence for sequence of fuzzy mappings.

Essentially, motivated by the works mentioned above, we introduced «^-statistically pointwise ideal convergence, «^-statistically uniform ideal convergence, and aft-equi-statistically ideal convergence for sequences of fuzzy-valued functions. With the help of certain examples, we present some relations between these type of convergence. Moreover, from an application point of view, we establish a new form of aft-statistical ideal Egorov's theorem for sequences of fuzzy-valued measurable functions defined on (Z, A,

A fuzzy set is a mapping Q: M ^ [0,1] that fulfill the following requirements:

(i) Q is normal,

(ii) Q is fuzzy convex,

(iii) Q is upper semi-continuous,

(iv) supp Q = cl{u e M: Q(u) > 0} is compact.

We represent by F(M) the set of all fuzzy numbers. The set M is involved in F(M) if we consider t e F(M) as

1, if u = t,

t(u) = ^ '

I supp Q, if u ^ t.

For a e (0,1], the a—cut of Q is given by [Q]a = {u e M: Q(u) ^ a}.

The Hausdorff distance between Q and q is denoted by D: F(M) x F(M) ^ [0, 8] and is given as

D(Q,q) = sup d([Q]a, [q]a) = sup max{|Qa — Q+1, \q~ — q+\},

ae[0,1] ae[0,1]

where d represents the Hausdorff metric. To learn more about fuzzy sequences, see [5] and [13]. For T c N, the natural density of T is given by

5(T) = lim 1 \{k ^ s: k e T}\,

S

if the limit exists; the vertical bars above denote the cardinality of the set. A sequence z = (zs) is statistically convergent to z if

¿({s e N: \zs — z\ ^ e}) = 0

for every e > 0.

Let Z be a non-empty set. Then a family of sets I C 2Z (the power set of Z) is said to be an ideal on Z iff

(i) 0 e I,

(ii) I is additive, that is, U,V e I ^ U y V e I,

(iii) U e I,V C U ^ V e I.

A non-empty family of sets F C 2Z is said to be a filter on Z if and only if $ R F. For U,V e F, we have U x V e F, and for each U e F and U C V this implies V e F. An ideal I c 2Z is called non-trivial if I ^ 2Z. There is a filter F(I) = {K C z: Kc e I}, for each I, where Kc = Z — K. A non-trivial I C 2Z is called admissible if {{z}: z e Z} C /. A non-trivial ideal is maximal if there cannot exist any non-trivial ideal J ^ I containing I as a subset. A non-trivial ideal I is called translation invariant ideal if for any U e I the set {k + 1: k e U} e I.

A sequence z = (zk) is said to I—convergent to the number z, if Ve > 0:

{k e N: \zfc — z\ ^ e} e I.

A function P(N) ^ [0, 8) (where P(N) is the power set of N) is a submeasure on N if

(i) W) = 0,

(ii) V(U) ^ V(U Y V) ^ V(U + V) V U,V c N,

(iii) ^ is lower semicontinuous if V U c N, ) = lim x s). Let || • : P(N) ^ [0, 8) be the submeasure defined by

||* = lim sup \s) = lim ^(U\s).

Consider Exh(^) = {U C N: = 0}. Therefore, Exh(^) is an ideal

for any submeasure

Remark. Let us choose I = Is = {T c N: 8(T) = 0}, where 8(T) represents the asymptotic density of set T; then Is is a non-trivial admissible ideal of N and the corresponding convergence coincides with the statistical convergence.

The concept of aft-statistical convergence was given by Aktuglu [1]. Assume that a(s) and ft(s) are two sequences of positive numbers, which fulfill the following conditions:

(i) a and ft are both non-decreasing,

(ii) ft(S) ^ a(s),

(iii) ft(s) — a(s) —> 8 as s — 8.

The set of pairs a,ft satisfying (i), (ii), and (iii) will be denoted A. For each pair (a, ft) e A and T c N, we define the density 8a"3(T) as:

rp pO. 3

^(T) = lim —-f^-, (C3 = [a(s),ft(s)]).

ft(s) — a(s) + 1

A sequence z = (zn) is said to be aft-statistically convergent to z, if for

\k e Ca/31

¿«'ß({fc 6 C«'ß: \za — z\ > e}) = lim 1

ft(s) — a(s) + 1

= 0,

for every e > 0; it is denoted by Saß.

Throughout the text, we denote a sequence of fuzzy-valued functions by SFVF and a fuzzy-valued function by FVF.

Definition 1. A SFVF (hv) is aft-statistically pointwise ideal convergent to FVF h on \a, b], if for every z 6 \a, b], i. e., V e > 0, 8 > 0, £ > 0, V z 6 \a, b]

|r 6 N : s 6 N:

1

ft (s) — a(s) + 1

{u 6 C«'ß: D(hv(z),h(z)) > ^ ^\r) ^ e}

belongs to I. In other words, Vz 6 \a, b], Ve,8 > 0, V£ > 0 D Tz 6 I s. t.,

m Ue N:

1

ft (s) — a(s) + 1

{u 6 C«'ß: D(K(z),h(z)) ^ e}| ^ ^\r) < £

x

X

X

X

Saß P1 q

V r e N\TZ. We write hu —-> h on [a, b] or Saß —pointwise ideal convergence.

Definition 2. A SFVF (hv) is aß-statistically uniform ideal convergent to FVF h on [a,b], i.e., V e > 0,5 > 0,f > 0, V z e [a,b]

|r e N : s e N:

1

ß(s) — a(s) + 1

[u e C:,ß: D(hv(z),h(z)) > e}| ^ \f) ^ 4

belongs to I. In other words, Vz e [a,b], Ve, 8 > 0, V£ > 0 D T e I s.t., V(ise N: 1

ß (s) — a(s) + 1

[u e C:,ß : D(hv(z),h(z)) > ^ | ^ V) < C

Vre N\ T. We write K

sUaB pi q

h on [a, b] or Saß-uniform ideal convergence.

Definition 3. A SFVF (hv) is aß-equi-statistically ideal convergent to FVF h on [a, b], iff for all e > 0,8 > 0, £ > 0, s. t.,

|r e N e N: 1

ß(s) — a(s) + 1

v e Ca/: D(hv(z),h(z)) ^ e} | ^ \r) ^ c}

w.r.t., z e [a, b] is uniformly convergent to zero function. It is denoted by hv ^ S^ß(I)h or Saß-equi ideally convergence.

2. Relation Between different convergence concepts of sequences of fuzzy-valued functions.

sUaB piq

€B pi q

P ' S^ßp /

Remark 1. If hv ^ h, then hv ——->• h. The converse is not neces-

sarily true.

Let us show this by an example.

Example 1. For any z e [0,1], let

™ ;), for z e [0,1];

K (z) = <

_ 1 + V2 z2, 0, otherwise.

x

X

X

X

Then (hu) is «^-statistically pointwise ideal convergent to h. However, for v e Cf:

- vz 1 1 sup D(h„(z),0) = sup --— = sup i-= - ^ 0.

.ze[0,l] 2:e[0,l] 1 + v % ze[0,1] ~ + VZ 2

Therefore, (hu) is not aft-statistically uniformly convergent nor aft-statistically uniformly ideal convergent to 0 on [0,1].

Corollary. Let (hu) be a SFVF and h be a FVF on [a,b]. Then

hv

suaB (i)

h ^ hv ^ (!)h ^ h

sZ0 (i)

h,

but converse is not true in general.

s? (I)

—-> h is not true, let

To show that the converse of hv ^ Seajj(I)h ^ hu us consider an example.

Example 2. Consider the SFVFs (hu) defined by hv(z) = e"uz for z e [0,1]. Then D(h„(z), 0) = e"^ for z e [0,1] for each v e N. Therefore, (hv) is aft-statistically pointwise ideal convergent to h = 0. However, for each r e N, consider r e [u, 2u — 1]. Therefore, for all z e [u, 2u — 1], we get

D(hr(z), 0) = e~vz ^ e-(2u~1)z ^ - ^ - (as z e

Thus, for all z e [0, ], we get

0,

2v- 1J

{r e [u, 2u — 1]: D(hr(z), 0) ^ 3} | ^ 0)

ft (s) — a(s) + 1

so, hv(z) is not aft-equi-statistically ideal convergent to FVF 0 on [0,1].

To show that the converse of hu let us consider an example.

Example 3. Suppose

sUaP (i)

h ^ hu ^ Seajj(I)h is not true,

hv (z) = <

1 — z2(p + 1)2

1 + z2

for z e

0,

v + 1

0,

otherwise.

u

1

1

1

Then

1

ß(s)- a(s) + 1

{u e Cf : D(hv(z), 0) ^ e}

1

ß (s)- a(s) + 1

0

as s ^ 8. Then (hv) is aß-equi-statistically ideal convergent to h = 0. However, for all u e C™13:

- 1 - z2(s + 1)2 sup D(hu(z), 0) = sup ---2- ^ 0,

,ze[0,1] -ze[0,1] 1 + z

for all v e C"13. Hence, (hv) is not aß-statistically uniformly ideally convergence to 0 on [0,1].

Proposition. Saß-equi ideally convergence is defined for j, such that 0 ^ 7 ^ 1.

Proof. Let us prove this result by contradiction. Consider two lower semicontinuous submeasures and such that Exh(^1) = Exh(^2) and @ £1 > 0,^1 > 0, D r e N, @ z e Z,

s e N

1

ß (s)- a(s) + 1 x

e Ca/ : D(hv (z),h(z )) ^ e^ ¿1} \r) < £1, @ e2 > 0,£2 > 0, D r e N, @ z e Z,

se N

1

ß (s)- a(s) + 1 x

e C^ : D(hv (z),h(z )) ^ e^ ¿2} \r) < £2. For all z e Z and for 7 e [0,1], we take

p(z) = e N :

= \s e N :

1

ß (s)- a(s) + 1

u e Ca/ : D(hu(z),h(z)) ^ £2}

> ¿2 =

1

ß (s)- a(s) + 1

e Casß : sup max) (hv (z))7 - (h(z))J

7e[0,1]

(K(z))+ - (h(z)) + \ } ^ 62} ¿2}

—>

X

X

X

X

and

pi(z) = {s e N:

1

ft (s) — a(s) + 1

ueC

a,[3 .

(hu (Z))z — (h(z)):

> £2

} ^ 4

P2(Z) = {

= \s e N:

1

ft (s) — a(s) + 1 Now, we take r1 and z1 s.t.,

e C*/: | (hu(z))+( —(h(z))+ 62} ¿2}.

^l(Pl(Zi)\n) < 2 , ^i(P2 (¿l)Vl) < 2

and

^2(Pl(^l)\rl) ^ £2, *2(P2(Zl)Vl) ^ £2. Since submeasure is lower continuous, we get r[ with

^2((Pl(Zl)Vl) X r'l) ^ |,

^2((p2(^l)\rl) X f'l) ^ |. Suppose that we have found r'm. Choose rm+1 > r'm and zm+1, such that

^l(Pl(*m+ l)Vm+l) < —1T, ^l(p2^m+l)Vm+l) < 1

2'+1

2'+1

and

^2(Pl(^m+ l)Vm+l) ^ £2, ^2(p2(zm+ l)\rm+1) ^ £2. From the lower continuity of we have r'm+1 with

-Till W \ 1 \ ^ £2

^2((Pl (Zm+1 AW l) X rm+1) ^ —, ^2((P2(Zm+1 Awl) X r'm+1) ^ ^.

We put Q = (pi(zm)\rm) x r'm)j , R = ume^(zm)\rm) x r'^.

Then we get

= lim ^i(Q\rm) ^

8

^ lim V ^1(P1(Zm) x {rm,r'm}) <

m

m^8

s=m

<

8 i

lim V — = 0,

m

m^8 ¿—L 2s s=m

and

||R||= lim R\rm) ^

m^8

8

^ lim V ^1(P2(Zm) x {^r'm}) <

s=m

<

8 i

lim V — = 0.

m

m^8 ¿—L 2

s=m

Thus, we have Q e Exh(^1) and R e Exh(^1). On the other hand, we have

||Q||*2 = lim ^2(QVm) ^ ^ > 0

and

||R||*2 = lim *2(RVm) > ^ > 0, which is a contradiction. □

Theorem 1. Consider an admissible ideal I. Let ( hv) be a SFVF and

Sat3 CO

h be a FVF, defined on [a, b]. For every z e [a, b], if hv(z) -> h(z),

sp„ pi)

then [hv(z)]7 [h(z)]7

w.r.t J.

S^s (-0

Proof. Assume that for every z e [a, b], hv(z) -> h(z). Let e > 0 be

given. Then, for each z e Z, there exists an integer i = i(z, e) e N s.t., V e > 0,5 > 0, V£ > 0,

< (1)

for all r ^ i. Thus, for any 7 e [0,1], Ve, > 0, 1

s e N:

/3(s) - a(s) + 1

ueC

a,/ ,

sup max (hv(z))y - (h(z))-'(=m n I '

7e[0,1]

(hv(z)) + -(^))7+} e} s}\r) < £ (2)

for all r ^ i.

For each 7 e [0,1], Ve,8,£ > 0,

m \se N:

1

P(s) - a(s) + 1

v e

QOi,fi :

(hv(z))~ -(h(z))- e)¿}\r) < £ (3)

for all r ^ i. Also, Ve, 5,^ > 0, 1

m ^ N:

P(s) - a(s) + 1

u e C^:

for all r ^ i From (1), we get Ve, > 0, r e N: s e N: 1

(hv(z)) + -(^))+ e)¿}\r) < £ (4)

P(s) - a(s) + 1

X IV e C^: D(hv (z),h(z)) ^ e} ^ \r) ^ ^ c N y {1, 2,...,«-1}. From (2), we get, for all Ve,5,£ > 0:

!

reN: m U e N:

1

P (s)-a(s) + 1

Qa,/3

sup max 7e[0,1]

( hu(z))- - ( h(z)):

( hv(z)) + - ( h(z)) + | } ^ e} | ^ S}v) ^ e} c N Y {1, 2, for any 7 e [0,1]. Equation (3) and (4) imply

,i - 1}.

x

X

X

X

X

reN:mU eN:

1

P(s)-a(s) + 1

{u e C^ : | ( hv(z))~-( h(z))-| ^ e|| ^ S}\r) } c Nu{1, 2,..., ¿-1}

and

reN:mU e N:

P (s)-a(s) + 1 ( K (z))7f -( h(z))+ e) | ^ s}\r) ^ e} c Ny{1, 2,..., ¿-1}.

As I is an admissible ideal, we get N y{1, 2,... ,i - 1} e I and, hence, for each e > 0 and for every £ > 0,

|r e N: s e N:

1

P (s)-a(s) + 1

v e : | ( hv (z))~ -( h(z))~ }\r) ^

and

|r e N: s e N:

P (s)-a(s) + 1 f e C3 : | ( K(z))7( - ( h( z))7 | ^ e} | ^ ¿}\r) ^

.

Sa/3 CO

Thus, for every z e [a, b] we get [h^(z)]7-> [h(z)]7 w.r.t. 7. □

Before starting Egorov's theorem, let us choose a finite measurable set Z. Take E c N (finite), consider [E, s] = {G c N: G x s = E}. The [E, s] is a base for the Cantor set on P(N).

3. aP-statistical ideal version of Egorov's theorem for the sequences of fuzzy-valued measurable functions.

Theorem 2. Let ( Z, A, p) be a finite measure space and I be an analytic P-ideal. Suppose that FVF h and SFVF (hv) are measurable and

S? (I)

defined on a.e. on Z. Also, assume that [hu(z)]7 —--> [h(z)]7 almost

everywhere on Z. Then, for each e > 0 D E c Z, such that p(Z\E) < e

S^ (I)

and hv\E —--> h\E on E.

X

1

X

1

Proof. Consider a finite measure space ( Z, A, y). Suppose that V m e N, h, and (hu) are defined everywhere on Z. For any fixed r,5,£ e N, consider the set

{z e Z: mQi

= \ z e Z: m < se N:

1

P (s)-a(s) + 1

{u e C?*: D(hv(z),h(z)) > ^ \r) < .

To show that Fr,s^ is measurable, we need to prove that the complement of every Fr,s^ is measurable.

X |i/ e Cf'P: D(hv(z),h(z)) > 1}

> -^j\rj e (1, 8

As m is lower semicontinuous, D set [Em, sm], such that

Z\Fr,S,i =

{u e C?* :D(hv(z),h(z)) > 1}

Z : N :

P (s)-a(s) + 1

■}v) e U [ ETO, sm]}

^ 1.

meN j=r

UHfeZ: s e N

meN

1

P (s)-a(s) + 1

{ve C^: D(hv(z),h(z)) ^

1

> -

} !> 1)))

XBm (j)

Since ( hv) and h are measurable, the right-hand side of the equation above is measurable and, so, Z\Fr>s^ is measurable. For every £ e N, we have

co

Fr,s,a c Fr+n,s,i:, Z = U F^. Thus, y(Z) = lims^cv(FrM). Let e > 0

r=1

be given. For every £ e N, assume r(£) e N be s.t., y(Z\Frp^qtst^) < —j.

co co

Consider Eo = |J (Z\Fr(?)A?). So, we have y(Eo) ^ X! v{Z\Fr(t),s,z) < e.

co

Let E = Z\E0 = p| Fr(^),s^. Thus, y(Z\E) = y(E0)< e. So, we have, for

x

x

all £ > 0, 5> 0, Dr(0 e N, @ z e E,

*({'e n : Pis,-e : D(h- w»*1 (| >1 ^ < i

This proves that hv\E is aP-statistically equi-ideal convergent to h\E on E. □ \ \

4. Conclusion. Upon the preceding analysis, our interest is to modify the studies of Kisi [10] nd investigate aP-statistical pointwise ideal convergence, aP-statistical uniform ideal convergence, aP-equi-statistically ideal convergence for sequences of FVFs. We demonstrate aP-statistically ideal version of Egorov's theorem for sequences of fuzzy-valued measurable functions on ( Z, A,p). As the future work, we are going to investigate Korovkin-type approximation theorems using aP-statistically ideal convergence for double sequences.

Acknowledgment. The authors deeply appreciate the suggestions of the reviewers and the editor that improved the paper.

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Received May 26, 2022. In revised form,, October 06, 2022.

Accepted October 12, 2022. Published online November 16, 2022.

School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J&K, India

Sonali Sharma

E-mail: [email protected] Kuldip Raj

E-mail: [email protected]