Statistical Bochner Integral on Frechet Space

ANITA CAUSHI1,2, ERVENILA MUSTA1,2

1Department of Mathematics, Polytechnic University of Tirana,

Mother Teresa Square 4, Tirana,

ALBANIA

2Department of Mathematical Engineering,

Faculty of Mathematics Engineering and Physical Engineering, Polytechnic University of Tirana,

ALBANIA

Abstract: - Probability theory including the Bochner integral is a very important part of modern mathematical

concepts including the modern theory of probabilities, especially in the concept of mathematical expectation

and dispersion. In this, study a statistical approach form of Bochner’s theory is given and extended, but some

fundamental properties of statistical integral were previously studied in the Banach case. Our approach

formulates an extended integration concept of Bochner. By using the statistical convergence on general locally

convex space it is possible to obtain very similar results referring to the Frechet space type. From our results,

some interesting comparable outputs to Banach space are carried out. At the end of our research, it is conducted

that if a function “f” is Bochner integrable in the classic report then it is statistically Bochner integrable, but

conversely, this is not true. Hence, the value of the extension of Bochner integration is a need and is the focus

of our work. This extension is given by modifying the model published by Schvabik and Guoju.

Mathematically it is substantiated that on the space of Frechet types the space of functions of statistical

Bochner integrable is a Frechet space.

Key-Words: - Statistical convergence, st-measurability, st-Bochner integral, statistical Frechet space,st-Cauchy

Convergence, st-strong measurable.

Received: July 22, 2022. Revised: January 21, 2023. Accepted: February 17, 2023. Published: March 30, 2023.

1 Introduction

The subject of statistical convergence has been

studied by many researchers since the emergence of

the idea of statistical convergence in 1935. Based on

statistical convergence a lot of effort is done by

many researchers and, [1], on the concept of

statistical Bochner integral. The problem of

determining a suitable representation for a fractional

power of an operator defined in a Banach space X

has, in recent years, attracted much attention even in

the solution of many engineering problems. A

perfect integration theory based on the concept of

Riemann-type integral sums was initiated around

1960 by Jaroslav Kurzweil and independently by

Ralph Henstock. Much of this theory is presented

and extended in the growing number of universities`

facts about the Henstock-Kurzweil integral known

as the generalized Riemann integral. The relatively

new concepts of the Henstock-Kurzweil and

McShane integrals based on Riemann-type sums are

an interesting challenge also in the study of the

integration of function on Banach space value.

Further on, the idea of statistical convergence was

introduced by Zigmund, [3]. The concept was

formalized by Steinhaus, [4], while the concept of

statistical convergence of sequences in real numbers

was introduced in 1951 by Fast, [5]. Later, the

concept was reintroduced by Schoenberg, [6]. The

concept of statistical convergence has become an

active area of research in recent years. Statistical

analogues of limit point results are obtained by

Fridy, [7], defining a statistical limit point of x as a

number  that is the limit of a subsequence 󰇝󰇛󰇜󰇞

of x such that the set 󰇝󰇛󰇜: ∈ 󰇞 does not have

density zero and the convergence of sequences and

the concept of Schoenberg about integration. The

basic concept of Fridy’s is the statistical Cauchy

convergence, [8]. On the other hand, we are affected

by the work performed by Connor, [9], which gives

proof that if a space does not contain a copy of l1,

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then every bounded weakly statistically null

sequence contains a weakly null subsequence, it is

helpful to recall the following result of Rosenthal,

[10]. In the study performed by [11], a contribution

to the theory of divergent sequences is defined and

examined by applying a new method of summation

which assigns a general limit , to certain

bounded sequences 󰇛󰇜. This method is

analogous to the mean values which are used in the

theory of almost periodic functions, furthermore, it

is narrowly connected with the limits of S. Lorentz

defined the space f of almost convergent sequences,

using the idea of Banach limits.

In the study published by authors in [12], a

characterization of statistical convergence of

sequences in topological groups is obtained and

extensions of a decomposition theorem, a

completeness theorem, and a Tauberian theorem are

given in [13], to the topological group setting are

proved. According to [14], a convergence of double

sequence spaces in 2-normed spaces and obtained a

criterion for double sequences in 2-normed spaces

to be statistically Cauchy sequence in 2-normed

spaces is given. While in the study given by [15],

various kinds of statistical convergence and -

convergence for sequences of functions with values

in R or, a metric space is discussed. For real-valued

measurable functions defined on a measure space

󰇛, , 󰇜 it is obtained a statistical version of the

Egorov theorem (when μ(X)<∞), a classical result

of measure theory. In the study of Maddox, [16], a

very interesting way of statistical approach is given.

In the study performed by authors in [17], very

interesting results presented are easily obtained

using a generalized version of the Bochner

technique due to theorems on the connection

between the geometry of a complete Riemannian

manifold and the global behavior of its

subharmonic, superharmonic, and convex functions.

Hence, the application of such old methods is

proven to have a great impact on engineering issues.

Based on the above theories our paper addresses the

issue of st-measurability for functions with values in

a Frechet space and relates it to the st-measurability

of functions in a Banach space. The novelty of our

study demonstrates that the essential properties of

st-measurability for functions in Banach spaces have

corresponding properties for functions with values

in Frechet spaces. Furthermore, the study establishes

several fundamental properties of the st-Bochner

integral in Frechet spaces, which are like those of

the st-Bochner integral for functions with values in

Banach spaces. Additionally, the paper shows that

the set of st-Bochner integrable functions is st-

Frechet.

2 Preliminary Approach Used

Let be  a subset of the ordered natural set N. It is

said to have density 󰇛󰇜 if 󰇛󰇜lim

→

||

 ,

where

󰇝:

󰇞 and with |A| denotes the

cardinality of set A. The finite sets have the density

zero and δ(A)= 1-δ(A) if A'=-A. If a property

Q(k)={k: kA} holds for all kA with δ(A)=1, we

say that property Q holds for almost all k that is

a.a.k. Let (X,p) be a semi-normed space, the

vectorial sequence 󰇛

󰇜 is statistically

convergent to the vector(element) L if for each ε >0



lim | : p(x ) | 0

kn L









i.e. p(xk –L)< a.a.k. We write st-limxk =L.

Lemma1, [18]:

A sequence 󰇛

󰇜is statistically convergent to L

if and only if there exists a set K={k1<k2<…}

that (K)=1 and lim( )

 

The set K is directed and the sequence ()

called the essential subsequence of (xk). The above

lemma can be formulated: A sequence (xk) is

statistically convergent to L if and only if there

exists an essential subsequence that converges in

usual meaning to limes L. We write now

lim k



we deal with the generalization of

pointwise statistical convergence of functions on

semi-normed space.

Definition 2.

A sequence of functions {fk(x)} is said to be

pointwise statistically convergent to f if for every

>0

lim |{ : p(f ( ) ( )) , }| 0

nkn x fx xS









i.e., for every xS, p(fk(x)-f(x))< all allmost k. We

write   

󰇛󰇜

󰇛󰇜or st

f on S. This

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means that for every δ>0, there exists integer N such

that:

lim |{ : p(f ( ) ( )) , }|

nkn x fx xS





 

For all n>N = (N(ε,δ,x)) and for every ε>0.

We can formulate an immediate corollary of Salat’s

lemma, [18]. The sequence {fk(x)} where fn:SX,

(X-a vectorial normed space) is statistically

convergent to f(x), if and only if, there exists an

essential subsequence ()

of it that is convergent

to f(x).

Following the definition of Cauchy sequences

introduced by Fridy, [7], and their extension to the

functional sequences and from the study performed

by authors, [18], the sequence (fk) is called the

statistically Cauchy (or st-Caushy) sequence if, for

every ε>0, there exists an integer N(=N(ε,x)) with:

lim |{ : p(f ( ) ( )) x S}|=0

nkN

kn x fx



 

A vectorial space V is statistically complete if each

st-Caushy sequence is convergent by the metric of

this space. Further, we denote (S,∑,μ) the

probability complete measure space, where S is any

set and  sigma -algebra of Borel.

Definition 3. A function f: S X, where X is a

vectorial semi-normed space is called a simple

function by μ, if there are finite sequence

measurable sets {Ei}, such that Ei S, i=1,...,n,

EiEj =  for ij, S = 1



 and f(s) = xi for s

Ei, It represented in a form = 1i





, where



is a characteristic function of Ei.

Let (X,p) be a semi-normed space, the function

f: SX is called st- Bochner integrable if there

exists st- Cauchy sequence of simple functions (fk)

such that :

i) statistically convergent a.e. by to the

function f.

ii) lim ( ( ) ( )) 0

st p f s f s d







st-lim ()



 is called st- Bochner integral and

denoted with (sB) ()



.

If the function f is Bochner integrable in the classic

definition then it is statistically Bochner integrable,

but conversely is not true. This gives the value of

the extension of Bochner integration in our article.

Let us show this by the modification of one example

published by Schvabik and Guoju, [2].

Example 4. Let f : [0,1]X be the function

11 ,

], ]

122

(0) 0

kk k

fzf











where X is a Banach space and (zk) is a sequence of

elements belonging to X and 1

|| || , 1

2kX

kzBB

It is proved that the function f is Bochner integrable

if and only if the series is convergent and

lim ( )

zMf

 







. By choosing this form

2 for k = n

1 for others

z





then the limes

lim 2

 



does not exist and the

function is not Bochner integrable, but this does not

prevent the function to be statistically Bochner

integrable.

3 Main Results and Discussion

A vectorial space V is called statistically Frechet

(shortly st-Frechet) if it is Hausdorf space,

statistically complete and its topology is inducted by

a countable family of semi-norms (pk).

Definition 5. Let (V,(pk)) be a statistically Frechet

space and the function f : SV. Then the function f

is measurable by  on (V, (pk)) if there exists a

sequence of simple functions (fn) convergent almost

everywhere to function f.

Lemma 6. [18]: Let (S, , ) be a measurable space

and f: SV is a statistically measurable function by

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 on the component (V, pk) of the Frechet space (V,

(pk)). Then there exists a finite or infinite sequence

of the disjoint sets (Sr) and

1) (S\(S1…Sr)< 1

2) The set f(Sr) is totally bounded on

space(V,pk).

We can represent the Frechet space only from

measurability in every component.

Theorem 7. Let (V,(pk)) be a st-Frechet space and

the function f :S  V. This function is measurable

by  on (V,(pk)) if and only if it is measurable by

each component (V, pk) of the space (V,(pk)).

Proof. The first part of the proof is derived from the

definition of measurability. Let’s use the sequence

(Tm) defined for the lemma 6, and fix the natural

number k. Since the set f(Tm) is totally bounded

(Lemma 6) then from one open cover 1

(,())

sof

this set one finds its finite subcover 1

(,())

krp

Rfs=.

Set 1

1[,()]

kkr

ij i

FTfRfs

=ΗΞSand further

11221

kkkkk

FE F F== …

\( ... )

kkk k

ii i

FF F

=ΘΘ .

The sets k

are disjoint family and 1

It holds furthermore then

( ) ( i {1,2,..., }, ( ( ) ( ) )

mki

sT ppfx fs"Ξ$Ξ-<

Set the sequence of the simple functions.

() 0.

ri ST

ffscc

then for every s  Tm

[() ()] 2

kr r

pfs fs-<

When r  the sequence of the simple functions

(fr) converges to f(x) on the arbitrary component (V,

pk). We prove that the function f(x) is measurable in

the space (V,(pk)).

Definition 8. The function f: S  V, (V, (pk) ) st-

Frechet space is called statistical Bohner integrable

(short sB- integrable) if there exists one sequence of

simple functions (fn) which converges almost

everywhere by measure  to the function f of this

space and for every continuous semi -norm pk holds.

lim ( () ()) 0

pfs fsd









The sequence (fn) is called the determinant of the

function f.

The following theorem proves that we find a

function that constructs the image of (Vpk) on

Freche space to the Banach space.

Theorem 9. Let (V, pk) be one component of the st-

Frechet space (V, (pk)), and the function f: S  V

is sB- integrable on (V, (pk)). Then for every pk, the

function '



is statistical Bohner

integrable on the subspace ( ''

Vp ) of the Banach

space ,

()

Proof. Let (fn) be one determinant sequence of the

simple functions of the function f. Since for every

pk and s  S, we can write

[() ()] '[ () ()]

kn k p n p

fs fs p fs fs







From this equation derives that the sequence of the

simple functions k



 converges almost

everywhere to the function k



 of the space

(''

Vp). The equation below shows that:

[ ( ) ( )] [ ( ) ( )]

kn k p n p

fs fsd p fs fsd









the sequence ( k



) is the determinant of the

function k



on the space ( ''

Vp) therefore this

function is statistically Bohner integrable.

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The function : →  is called statistically strong

measurable by  on set S , (in short form st-

measurable) if there exists a sequence of simple

functions 

 that for every ∈ and every 0

holds:

lim

→



∶

󰇛󰇜

󰇛󰇜  , ∀ ∈

0, for almost all ∈.

Theorem 9.(Theorem Egorov). If a function f:

S→X is st-measurable by μ, then it is st- strong

measurable uniformly almost everywhere on S.

The proof repeated the proof for the Banach space

done to Caushi, [1].

Theorem 10. Let X semi-normed space and the

function f: SX is statistically measurable. Then

1) The function f is with a separable value

almost everywhere by .

2) f-1(G) for every open set G.

Proof. First, show that the function f is almost

separable. The function f(x) is statistically

measurable therefore there exists the sequence of

simple functions (fn) and the set NS with (N)=0

such that:

(() ())

pf s fs





almost every n and sS\N.

Denote





:(() ())

Aknpfsfs



  .

By Egorov, Theorem 9, for every nAk there exists

a subset EnS with (En)< that the sequence of

simple functions (fn(s)) converges statistically and

uniformly to f(x) on S\En.

Since fn are simple functions its value fn(S) are finite

for every n. We know that the set ()



 is

countable and by considering this fact for each n,

( (\)) (\)

nA nA

SE fSE







then it is easily provided the mathematical

expression as below:

( (\)) (\)

nnn

nA nA

SE fSE







The set (\ )



 is separable as the closure

of the countable set.

If considering (En)< 1

n for nAk, then it can be

given such as:

ES SE

 for nAk

and

()()0

ESSE





 .

Putting N= n

for nAk we prove the separability

of the set {f(s),sS\N}.

(b) Let G be an open set in X. We denote

((),)Rfs n the set 1

{:(())}yXpyfs n



and

Gn=

()

((),)

fs G

Rfs n



. It is easy to prove as



If sS\Z then from the equality st-

lim () ()

sfs





we have that fk1

((),)Rfs n for

kK and k>m. So the f(s)G if and only if fkGn

k>m. Hence, the model can be reformulated by the

equation:

1, 1 ,

()\ ( ( )\)

nmkKkm

GZ f G Z









which proves that 1()

.

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The following lemma is a key lemma to prove that

the space (sB(m, V),(qk)) is statistically-Frechet.

Lemma 11. Let (fn) be a sequence st-Cauchy of

simple functions on the vectorial space (sB(,

V),(qk)). Then there exists a subsequence (gn) of (fn)

convergent to one function f of the Frechet space

(',( )

Vq).

Proof. Since the sequence (fn) is a Cauchy sequence

on the space (sB(, V),(qk)) then for every n  K

there exists 2

()

N such that for every n>N:

()

kn N r

qf f. By the Salat lemma, [11], we

can order the elements of the set K as r1 < r2<…<

rn<…

By denoting gm = m

, then

()()

kr r km N r

qf f qg g 

for every m> N

r . Set the series (1)

() [ () ()]

gs g s gs











for every sS. We note

Mr ={s S: 1

(() ())

ki i r

pg s gs



}

and

() (() ())

rkii

dpgsgsd







 



[() ()]

ki i r

qg s gs





Hence,

()



 .

Putting

Zr = Mr Mr+1 . . .

and it is verified that Zr+1  Z

r. But, on the other

hand, it is known that:

() ( ) 22

ir ir













and if the set r







then (Z) = 0. Let‘s be

an element such that s  S \ Z. Then there exists the

set 0

that 0



and for r  r0

[() ()] 2

ki i r

ir ir

pg x gx











because the seminorms (pk) is monotone non-

decreasing and for i  k

pk[gi+1(s) – gi(s)]pi[gi+1(s)-gi(s)] < 1

2i.

as a consequence the series

[ ( )] [ ( ) ( )]

kkii

pgs pg s gs











is convergent and this implies that series (1) is

convergent to the function f on the component (V,

pk) of the Frechet space (V, (pk)) for k  k0 and s 

S \ Z. Thus

lim ( ) lim p

sff



.

For every p  P, we set the function

:(,)(,), q() (()

pkp

qsBV X Vp f pfxd







It is easy to see that qp is a semi-norm on sBV(,V)

and this implies that the family of (qp)pP determines

one locally convex space.

In the following theorem, we prove the (sBV(, V),

(qk)) is statistically Frechet.

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Theorem 12. The vectorial topological space

(sBV(, V),(qk)) is statistically Frechet.

Proof. Let us show that the sequence of semi-norms

(qk) separates the points. Let f be a function of this

space and for every k 

() (() 0

qf pfsd







then there exists the set Zk such that (Zk)=0 for

which pk(f(s))=0 for the all sZk. If one takes





 then for every sZ and k we

obtain that p(f(s)=0. Since the family (pk) separates

the points then f=0 almost everywhere by  and so

are the seminorms qk.

Let us demonstrate that locally convex space

(sBV(, V),(qk)) is statistically Frechet space and let

(gn) be a statistically Cauchy sequence in that space.

By the definition of statistical Bohner integral, for

every n  K, there exists a simple function fn such

that:

qn (gn –fn)< 1

Let’s consider the sequential Cauchy sequence (gn)

for n > N then

()()

kn N kn n

kn N kN N

qf f qf g

qg g qg f

 

 

<11

()

kn N

pg g



So the sequence (fn) is statistically Cauchy of simple

functions in arbitrary component (sBV(, V),qk) of

the space (sBV(, V),(qk)). This implies the

sequence (fn) is statistically Cauchy in the space

(sBV(, V),(qk)). By the virtue of Lemma 11, the

sequence (fn) has one subsequence ( k

) converges

almost everywhere to the function f: SV of the

Frechet space (V, (pk). On the other hand, the

sequence ( k

) is a statistically also Cauchy

sequence in the space (sBV(, V),(qk).

The inequalities

()( )()

kkkk

kn kn n kn

qg f qg f qf f





show that the subsequence ( k

g) of the sequence

(gn) converges on every component (sBV( ,V), qk)

of the space (sBV(, V), (qk)). So the sequence (gn)

converges to f on the space (sBV(, V),(qk)). This

space is statistically complete by the sequences or

statistically Frechet.

4 Conclusions

First, we find another application of statistical

Bochner integral in Freshet space in comparison

with Banach ones. Our study proves the same

results of the classic Bochner integral on Freshet

space for the statistic Bochner integrable in more

simple and constructive ways. our method on

statistical Bochner integral in Freshet space is

further explained and extended for each case. The

contribution of the methodology in comparison to

other researchers which are introduced in a

generalized form.

At the end of our research, it is conducted that if a

function “f” is Bochner integrable in the classic

report then it is statistically Bochner integrable, but

conversely, this comes not true, at all. Hence, the

value of the extension of Bochner integration is a

need and is the focus of our work. This extension is

given by modifying the model published by

Schvabik and Guoju, [2]. Mathematically it is

proved that on the space of Frechet types the space

of functions of statistical Bochner integrable is a

Frechet space.

5 Future work

By considering the concept of Ideal convergence,

which is closely related to statistical convergence, it

is possible to establish the concept of measurability

within the framework of ideals. This, in turn, allows

for the construction of a form of ideal Bochner

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.27

Anita Caushi, Ervenila Musta

E-ISSN: 2224-2880

230

Volume 22, 2023

integral for functions that have values in a Frechet

space. Probability theory including statistical

Bochner integral in Freshet space is a very

important part of nowadays mathematical concepts,

especially in the concept of mathematical

expectation and dispersion. Our future work will be

focused on the application of the results in real

engineering systems.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

-Anita Caushi: Conceptualization of the published

work, formulation, and evolution of overarching

research goals and aims. Data curation and

scrubbing data and maintaining research data

(including proofing and validation.

-Ervenila Musta: Formal analysis and Preparation,

creation, and presentation of the published work.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

There are no sources of funding for this research

work.

Conflict of Interest

The authors have no conflict of interest to declare

that is relevant to the content of this article.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.27

Anita Caushi, Ervenila Musta

E-ISSN: 2224-2880

231

Volume 22, 2023