Vitali Theorems in Non-Newtonian Sense and Non-Newtonian
Measurable Functions
OGUZ OGUR, ZEKIYE GUNES
Department of Mathematics,
Giresun University,
Giresun-Gure,
TURKEY
Abstract: In this paper, we first state the ν−Vitali theorems in the non-Newtonian sense. In the second part, we
give the definition of the non-Newtonian measurable function and the relation between ν−measurable and real
measurable functions. We also study some basic properties of ν−-measurable functions.
Key-Words: Non-Newtonian measurable set, non-Newtonian Vitali set, non-Newtonian measurable function
Received: March 23, 2024. Revised: August 25, 2024. Accepted: September 18, 2024. Published: October 16, 2024.
1 Introduction
Non-Newtonian calculus, which has found
applications in various fields such as engineering,
mathematics, finance, economics, medicine, and
biomedical sciences, was developed between
1967 and 1970 as an alternative to the classical
calculus of Newton and Leibniz, [1], [2]. The
foundational book titled Non-Newtonian Calculus,
which laid the groundwork for this alternative
calculus, was published in 1972 by [3]. The
concepts of derivative and integral in the context of
metacalculus were explored by [4], while geometric
calculus and its applications were examined in
[5]. The non-Newtonian Lebesgue measure for
non-Newtonian open sets was defined and studied
in[6]. Finally, the non-Newtonian measure for
closed non-Newtonian sets, along with some related
theorems, was defined and studied in [7]. For more
details see, [8], [9], [10], [11], [12], [13], [14], [15],
[16], [17], [18], [19], [20], [21], [22].
Let νbe a generator, which means νis a
one-to-one function whose domain is real numbers
and whose range is a subset Aof R. Let ˙p, ˙q∈A.
Then, ν−arithmetics are defined as follows;
ν−addition ˙p˙
+ ˙q=ν{ν−1( ˙p) + ν−1( ˙q)}
ν−subtraction ˙p˙
−˙q=ν{ν−1( ˙p)−ν−1( ˙q)}
ν−multiplicative ˙p˙
×˙q=ν{ν−1( ˙p)×ν−1( ˙q)}
ν−division
(ν−1( ˙q)= 0) ˙p˙
/ ˙q=ν{ν−1( ˙p)/ν−1( ˙q)}
ν−order ˙p˙
≤˙q⇔ν−1( ˙p)≤ν−1( ˙q)
Numbers with x˙
>˙
0are called ν−positive numbers,
and numbers with x˙
<˙
0are called ν−negative
numbers. The set of ν−integers is
Zν=Z(N) = . . . , ν(−2), ν(−1), ν(0), ν(1), ν(2), . . . .
The set Rν=R(N) = {ν(a) : a∈R}is called
the set of non-Newtonian real numbers.
The absolute non-Newtonian value of ˙a∈Ain
the subset A⊂Rνis denoted by |˙a|Nand define as
follows;
|˙a|ν=
˙a , ˙a˙
>ν(0)
ν(0) ,˙a=ν(0)
ν(0) ˙
−˙a , ˙a˙
<ν(0)
Accordingly,
√˙a2N
N=|˙a|N=ν|ν−1( ˙a)|
is written for each ˙uin the set A⊂Rν[3], [8].
Definition 1. The non-Newtonian outer measure of
a nonempty ν−bounded set Kis the largest lower
bound of the measures of all ν−bounded, ν−open
sets containing the set K. So it is defined by
m∗
NK=νinf
K⊂G{mNG}
[7].
Definition 2. The non-Newtonian interior measure of
a nonempty ν−bounded set Kis the smallest upper
bound of the measures of all ν−closed sets contained
in the set K. So it is defined by
m∗NK=νsup
F⊂K{mNF}
[7].
Theorem 1. Let be given a ν−bounded set K. If ∆is
aν−open set containing the set K, then we have the
following equation;
m∗
NK˙
+m∗NCK
∆=mN∆
[7].
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Definition 3. If the non-Newtonian interior and
exterior measure of a ν−bounded set Kare equal,
the set Kis called a non-Newtonian Lebesgue
measurable set, or simply the ν−measurable set, [7].
Theorem 2. If the set Kis the ν−measurable set in
Rν, then ν−1(K)is the measurable set in R, [7].
Theorem 3. Let be given a ν−bounded set E.
If the set E can be written as a combination of
finite or countably infinite sets of pairwise disjoint
ν−measurable Eksets, then Eis ν−measurable and
mNE=ν
k
mNEk
equality is satisfied, [23].
2 Main Results
2.1 Vitali Theorems
Definition 4. Let Kbe a set of ν−points and Ba
family of ν−closed intervals, none of which are single
points.If for every x∈Kpoint and for every ˙ϵ˙
>˙
0
there is a ν−closed b∈Binterval such that
x∈b, mNb˙
<˙ϵ,
then, the set Kis said to be contained by the family
Bin the ν−Vitali sense.
In other words, if every point of the set Klies in
arbitrarily small ν−closed intervals belonging to the
family B, then the set Kis covered by the family B
in the ν−Vitali sense.
If the set Kis contained by the Bin the ν−Vitali
sense, then the set ν−1(K)is also contained by a
family in the Vitali sense. Let Bis the family of
ν−closed sets bwhich do not consist of a single point
and let B1be the family of ν−1(b)closed sets that do
not consist of a single point. Then, for ∀x∈Kand
for each ˙ϵ˙
>˙
0, there is an ν−closed interval b∈B
such that
x∈b, mNb˙
<˙ϵ.
Then, we have
ν−1(x)∈ν−1(b)
and ν−1(b)∈B1since b∈Bso we get
ν−1(mN(b)) < ν−1( ˙ϵ)
ν−1ν{m(ν−1(b))}< ϵ
m(ν−1(b)) < ϵ (ϵ > 0)
⇒ν−1(K)Vitali
Theorem 4. If a ν−bounded set Kis covered by a
family of closed intervals Bin the ν−Vitali sense,
then it is possible to find a finite or countable family
of ν−closed intervals bkin the set Bsuch that
bk∩bi=∅(k=i)m∗
NK\
k
bk=˙
0.
Proof. Since the set Kis ν−bounded, the set ν−1(K)
is bounded and is covered by the family B1which
consist of closed intervals. By the Vitali’s theorem it
is possible to find a finite or countable closed interval
family ν−1(bk)in the set B1, such that
⇒ν−1(bk)∩ν−1(bi) = ∅(k=i)
m∗ν−1(K)\
k
ν−1(bk)= 0
⇒ν{ν−1(bk∩bi)}=ν{∅}(k=i)
m∗ν−1(K)\ν−1
k
bk= 0
⇒bk∩bi=∅(k=i)
νm∗ν−1K\
k
bk=ν(0)
⇒bk∩bi=∅(k=i)
m∗
NK\
k
bk=˙
0
which gives the proof.
Theorem 5. Under the hypotheses of Theorem 4,
for every ˙ϵ˙
>˙
0there is a finite system b1, b2, . . . , bn
consisting of pairwise disjoint ν−closed intervals of
the system Bsuch that
m∗
NK\
n
k
bk˙
<˙ϵ.
Proof. If the set Kis covered by a family of closed
intervals Bin the sense of ν−Vitali, then the set
ν−1(K)is also covered by a family of closed intervals
B1in the sense of Vitali. The B1system has a finite
ν−1(b1), ν−1(b2), . . . , ν−1(bn)
system of pairwise disjoint closed intervals. Thus we
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get
⇒m∗ν−1(K)\
n
k
ν−1(bk)< ϵ
⇒m∗ν−1(K)\ν−1n
k
bk< ϵ
⇒νm∗ν−1K\
n
k
bk< ν(ϵ)
⇒m∗
NK\
n
k
bk<˙ϵ
⇒ν−1m∗
NK\
n
k
bk< ν−1( ˙ϵ)
⇒m∗
NK\
n
k
bk˙
<˙ϵ
.
2.2 Measurable Functions
Definition 5. Let
fν:X⊂Rν→Rν
˙a→fν( ˙a).
If for ∀˙
β∈Rν, the set
A={˙a∈X:fν( ˙a)˙
>˙
β}
is ν−measurable, that is,
m∗
NA=m∗NA
then, the function fνis called a non-Newtonian
measurable function, or simply a ν−measurable
function.
Theorem 6. Let fν:X⊂Rν→Rνbe a function.
The following expressions are equivalent; for ∀˙
β∈
Rν
a) the set A˙
β={˙a∈X:fν( ˙a)˙
>˙
β}is the
ν−measurable set,
b) the set B˙
β={˙a∈X:fν( ˙a)˙
≤˙
β}is the
ν−measurable set,
c) the set C˙
β={˙a∈X:fν( ˙a)˙
≥˙
β}is the
ν−measurable set,
b) the set D˙
β={˙a∈X:fν( ˙a)˙
<˙
β}is the
ν−measurable set.
Proof. It is obvious that A˙
β=X\B˙
β,B˙
β=X\A˙
β.
(a)⇒(b):Since A˙
βis ν−measurable, its
complement, B˙
β, is also ν−measurable.
(b)⇒(a): Since B˙
βis ν−measurable, its
complement, A˙
β, is also ν−measurable.
Thus, we get (a)⇔(b).
Since C˙
β=X\D˙
βis D˙
β=X\C˙
β(c)⇔(d).
(a)⇒(c): For ∀˙
β∈Rν, let A˙
β={˙a∈X:
fν( ˙a)˙
>˙
β}be the ν−measurable set.
For every ˙m ν−positive integer, we have ˙
β˙
−˙
1
m∈Rν
since ˙
β∈Rνand ˙
1
m∈Rνand so A˙
β˙
−˙
1
m
is a
ν−measurable set.
Thus
A˙
β˙
−˙
1
m
={˙a∈X:fν( ˙a)˙
>˙
β˙
−˙
1
m}
and we get
∞
m=1
A˙
β˙
−˙
1
m
=
∞
m=1 ˙a∈X:fν( ˙a)˙
>˙
β˙
−˙
1
m
={˙a∈X:fν( ˙a)˙
≥˙
β}=C˙
β
is the ν−measurable set.
(c)⇒(a): For ∀˙
β∈Rν,C˙
β={˙a∈X:fν( ˙a)˙
≥˙
β}
be a ν−measurable set.
For every ˙m ν−positive integer, we have ˙
β˙
+˙
1
m∈Rν
since ˙
β∈Rνand ˙
1
m∈Rνand so C˙
β˙
+˙
1
m
is the
ν−measurable set.
Again
C˙
β˙
+˙
1
m
={˙a∈X:fν( ˙a)˙
≥˙
β˙
+˙
1
m}
and we get
∞
n=1
C˙
β˙
+˙
1
m
=
∞
m=1 ˙a∈X:fν( ˙a)˙
≥˙
β˙
+˙
1
m
={˙a∈X:fν( ˙a)˙
>˙
β}=A˙
β
is the ν−measurable set which gives (a)⇔(c).
Theorem 7. If
fν:X⊂Rν→Rν
˙a→fν( ˙a)
is a non-Newtonian measurable function, then
ν−1◦fν◦ν:ν−1(X)⊂R→R
a→(ν−1◦fν◦ν)(a)
is a measurable function.
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Proof. Since fνis a non-Newtonian measurable
function, we have
∀˙
β∈Rν,{˙a∈X:fν( ˙a)˙
>˙
β}is the ν−measurable
set. For ∀β∈R, the set
ν−1({˙a∈X:fν( ˙a)˙
>˙
β})
⇔ {ν−1( ˙a)∈ν−1(X) : fν( ˙a)˙
>˙
β}
⇔ {a∈ν−1(X) : ν−1(fν(ν(a))) > ν−1(˙
β)}
⇔ {a∈ν−1(X) : (ν−1◦fν◦ν)(a)> β}
is measurable. This completes the proof.
Example 1. The non-Newtonian constant function
fν:X⊂Rν→Rν
˙a→fν( ˙a) = ˙c, ˙c∈Rν
is a ν−measurable.
Proof. For ∀˙
β∈Rν, it can be shown that the set
{˙a∈X:fν( ˙a) = ˙c˙
>˙
β}
is ν−measurable.
(i) Let ˙
β˙
≥˙c. Then, the set
fν( ˙a)˙
>˙
β˙
≥˙c
{˙a∈X:fν( ˙a)˙
>˙
β}=∅
is ν−measurable.
(ii) Let ˙
β˙
<˙c. Thus, the set
{˙a∈X:fν( ˙a)˙
>˙
β}=X
is ν−measurable.
Then the non-Newtonian constant function is
ν−measurable.
Example 2. For ∀˙a∈Rν, the set {˙a∈X:fν(˙a) =
˙
β}is a-measurable if
fν:X⊂Rν→Rν
˙a→fν( ˙a)
is a non-Newtonian measurable function.
Proof. It is easy to see the following equality:
{˙a∈X:fν( ˙a) = ˙
β}
={˙a∈X:fν( ˙a)˙
≥˙
β}∩{˙a∈X:fν( ˙a)˙
≤˙
β}.
Since finite number of intersections of ν−measurable
sets are ν−measurable the proof is completed.
Definition 6. Given a set E, the ν−characteristic
function of Eis denoted by νχand defined by
νχE=˙
1, x ∈E
˙
0, x /∈E
Example 3. If Eis ν−measurable set, the the
function νχEis a ν−measurable function.
Proof. For ∀β∈Rνwe show that the set
x∈X:νχE(X)˙
>˙
βis a ν−measurable.
i) If ˙
β˙
<˙
0, then the set
x∈X:νχE(X)˙
>˙
β=X
is ν−measurable.
ii) Let ˙
0˙
≤˙
β˙
<˙
1. Then the set
x∈X:νχE(X)˙
>˙
β=E
is ν−measurable.
iii) Let ˙
β˙
≥˙
1.
x∈X:νχE(X)˙
>˙
β=∅
set is ν−measurable.
Hence, the function νχEis ν−measurable function
when the set Eis ν−measurable.
Theorem 8. If the function fνis ν−measurable
non-Newtonian real-valued function and ˙c∈Rν, then
the function ˙c˙
×fνis ν−measurable.
Proof. To show that the function ( ˙c˙
×fν)( ˙a) =
˙c˙
×fν( ˙a)is ν−measurable, the set
{˙a∈X: ( ˙c˙
×fν)( ˙a)˙
>˙
β}
must be shown to be ν−measurable.
i) If ˙c=˙
0, the set
{˙a∈X: ( ˙c˙
×fν)( ˙a)˙
>˙
β}=Xif ˙
β˙
<˙
0
is ν−measurable and the set
{˙a∈X: ( ˙c˙
×fν)( ˙a)˙
>˙
β}=∅if ˙
β˙
≥˙
0
is ν−measurable.
which shows that the ˙c˙
×fνfunction is ν−measurable.
ii) Let ˙c˙
>˙
0. We write
{˙a∈X: ˙c˙
×fν( ˙a)˙
>˙
β}={˙a∈X:fν( ˙a)˙
>˙
β˙
/ ˙c}.
Since ˙
β˙
/ ˙c∈Rνand the fνfunction is ν−measurable,
the set {˙a∈X: ˙c˙
×fν( ˙a)˙
>˙
β}is ν−measurable.
iii) Let ˙c˙
<˙
0. The set
{˙a∈X: ˙c˙
×fν( ˙a)˙
<˙
β}={˙a∈X:fν( ˙a)˙
<˙
β˙
/ ˙c}
is ν−measurable since ˙
β˙
/ ˙c∈Rνand the function fν
is ν−measurable.
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Theorem 9. If the function fνis ν−measurable,
then the non-Newtonian real-valued function f2N
νis
ν−measurable.
Proof.
i) If ˙
β˙
<˙
0then {˙a∈X: [fν( ˙a)]2N˙
>˙
β}=Xset is
ν−measurable.
ii) If ˙
β˙
≥˙
0then we have
{˙a∈X: [fν( ˙a)]2N˙
>˙
β}
={˙a∈X:|fν( ˙a)|N˙
>˙
β
N
}
={˙a∈X:fν( ˙a)˙
>˙
β
N
}∪{˙a∈X:fν( ˙a)˙
<˙
−˙
β
N
}
which shows the function f2N
νis ν−measurable.
Theorem 10. If non-Newtonian real-valued functions
fν, gνare ν−measurable, then the function fν˙
+gνis
ν−measurable.
Proof. Since the functions fν, gνare ν−measurable,
then we have ν−1◦fν◦ν, ν−1◦gν◦νare real-valued
measurable functions. Therefore, we get (ν−1◦fν◦
ν)+(ν−1◦gν◦ν)is measurable.
Thus, since
(ν−1◦fν◦ν)(a)+(ν−1◦gν◦ν)(a)
=ν−1(ν{(ν−1◦fν◦ν)(a)+(ν−1◦gν◦ν)(a)})
=ν−1(ν{ν−1(fν(ν(a))) + ν−1(gν(ν(a)))})
=ν−1(fν(ν(a)) ˙
+gν(ν(a)))
=ν−1((fν˙
+gν)(ν(a)))
= (ν−1◦(fν˙
+gν)◦ν)(a)
is a measurable function for ∀a∈ν−1(X), then the
function fν˙
+gνis ν−measurable.
Theorem 11. If fν, gνare ν−measurable, then the
function fν˙
×gνis ν−measurable.
Proof. If fνand gνare ν−measurable, then ν−1◦fν◦
ν, ν−1◦gν◦νare real-valued measurable functions.
Therefore, the (ν−1◦fν◦ν)×(ν−1◦gν◦ν)function
is measurable. Thus, since
(ν−1◦fν◦ν)(a)×(ν−1◦gν◦ν)(a)
=ν−1(ν{(ν−1◦fν◦ν(a)) ×(ν−1◦gν◦ν(a))})
=ν−1(ν{ν−1(fν(ν(a))) ×ν−1(gν(ν(a)))})
=ν−1(fν(ν(a)) ˙
×gν(ν(a)))
=ν−1((fν˙
×gν)(ν(a)))
= (ν−1◦(fν˙
×gν)◦ν)(a)
is a measurable function for ∀a∈ν−1(X), then the
function fν˙
×gνis ν−measurable.
Theorem 12. If the function fνis ν−measurable,
then the function |fν|Nis ν−measurable.
Proof.
i) If ˙
β˙
<˙
0, the set {˙a∈X:|fν( ˙a)|N˙
>˙
β}=Xis
ν−measurable.
ii) Let ˙
β˙
≥˙
0. Then the set {˙a∈X:|fν( ˙a)|N˙
>˙
β}is
ν−measurable since
{˙a∈X:|fν( ˙a)|N˙
>˙
β}
={˙a∈X:fν( ˙a)˙
>˙
β}∪{˙a∈X:fν( ˙a)˙
<˙
−˙
β}
is ν−measurable. This shows that the function |fν|N
is ν−measurable.
3 Conclusion
In this study, we first give the ν−Vitali theorems
in the non-Newtonian sense. In the second
part, we give the definition of the non-Newtonian
measurable function. Also, we show that a function
ν−measurable if and only if the function ν−1◦fν◦νis
a measurable function.This can be seen as the crucial
step in the definition of the Lebesgue integral in
the non-Newtonian sense. We also investigate some
basic properties of ν−measurable functions.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.66
Oguz Ogur, Zekiye Gunes
E-ISSN: 2224-2880
632
Volume 23, 2024
