# Non-Newtonian Comment of Lebesgue Measure in Real Numbers

Authors: Cenap Duyar; Birsen Sağır
Journal: Journal of Mathematics
 (2017)

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## Abstract

We would like to generalize to non-Newtonian real numbers the usual Lebesgue measure in real numbers. For this purpose, we introduce the Lebesgue measure on open and closed sets in non-Newtonian sense and examine their basic properties.


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## Body

## 1. Introduction

A generator is a one-to-one functionα, whose domain is R, the set of all real numbers, and whose range is a subset of R. Identity function and exponential function can be given as examples of generators. We denote by RN, called non-Newtonian real line, the range of generator α. Non-Newtonian arithmetic operations on RN are represented as follows [1–5]:α-addition  x+.y=αα-1x+α-1yα-subtraction  x-.y=αα-1x-α-1yα-multiplication  x×.y=αα-1x×α-1yα-division  x/.y=αα-1x/α-1yα-order  x<.yx≤.y⇔α-1x<α-1yα-1x≤α-1yA closed interval onRN is represented by (1)[.a,b].=x∈RN:a≤.x≤.b=x∈RN:α-1a≤α-1x≤.α-1b=αα-1a,α-1b,and similarly an open interval (.a,b). can be represented [2, 6]. Let E be a subset in RN. A point x0 is called an interior point of E if there exists an interval (.a,b). with x0∈(.a,b).⊂E. A set E is said to be open in RN if all of its points are interior points. The component interval (.a,b). of an open set G in RN is an open interval providing conditions (.a,b).⊂G, a∉G, and b∉G. It is known that any open set in R is written as any composition of the component intervals, and also it would be an easy detection that the same thing is valid in RN [6–8].In this work, the symbolN∑‍ in RN is used instead of the usual summation symbol ∑‍.This work is deviated mainly to establish a measure in Lebesgue’s sense onRN.The measurema,b of an open interval a,b in R is the length b-a of this interval.All proven properties here are the generalization of basic measurement properties known in real analysis. The readers can refer to the textbook [9] for these properties.


## 2. Main Results

Definition 1.
The measuremN(.a,b). of an open interval (.a,b). in RN is defined by (2)mN(.a,b).=αmα-1a,α-1b.One can restate the above relation as follows:(3)mN(.a,b).=αmα-1a,α-1b=αα-1b-α-1a=b-.a.Lemma 2.
If a finite number of pairwise disjoint open intervals(.ak,bk)., k=1,2,…,n, are contained in the open interval (.A,B)., then(4)N∑k=1nmN​.ak,bk​.≤.mN​.A,B​..Proof.
We may suppose thata1<.a2<.⋯<.an. Then, by the properties of the generator α and the definition non-Newtonian sum, we have(5)N∑k=1nmN​.ak,bk​.=N∑k=1nαmα-1ak,α-1bk=α∑k=1nmα-1ak,α-1bk≤αmα-1A,α-1B=αα-1B-α-1A=mN​.A,B​..This completes the proof.Corollary 3.
If an infinite number of pairwise disjoint open intervals(.ak,bk)., k=1,2,…, in RN, are contained in the open interval (.A,B)., then(6)N∑k=1∞mN​.ak,bk​.≤.mN​.A,B​..Definition 4.
The measuremNG of a nonvoid bounded open set G in RN is the sum of the measures of all its component intervals δk:(7)mNG=N∑kmNδk.Here it should be noted that(8)mNG=N∑kmN​.ak,bk​.=N∑kαmα-1ak,α-1bk=α∑kmα-1ak,α-1bk=∑kbk-.ak,where δk=(.ak,bk).. Also it should be noted that(9)G=⋃kδk=⋃k​.ak,bk​.=⋃kαα-1ak,α-1bk=α⋃kα-1ak,α-1bk,since the generator α is a one-to-one function, and here the intervals (10)α-1ak,α-1bk,⁡k=1,2,…,are the component intervals of α-1G.Theorem 5.
LetG1 and G2 be two bounded open sets in RN. If G1⊂G2, then mNG1≤mNG2.Proof.
We can writeG1=∪iδi and G2=∪kθk as a composition of the component intervals δi=(.ai,bi). and θk=(.ck,dk).. Since each α-1δi is in one and only one α-1θk, we have (11)mNG1=N∑imNδi=N∑iαmα-1ai,α-1bi=α∑imα-1ai,α-1bi=α∑imα-1δi=α∑k∑α-1δi⊂α-1θkmα-1δi=α∑k∑α-1δi⊂α-1θkmα-1ai,α-1bi≤α∑kmα-1dk-α-1ck=N∑kmNθk=mNG2.This completes the proof.Theorem 6.
If the bounded open setG⊂RN is the composition of at most numerable family of pairwise disjoint open sets Gk, then mNG=N∑kmNGk.Proof.
If eachGk is written as a composition Gk=∪iδik of the component intervals δik=(.aik,bik)., then(12)mNG=mN⋃kGk=mN⋃k⋃iδik=mN⋃k⋃i(.aik,bik).=mN⋃k⋃iαα-1aik,α-1bik=mNα⋃k⋃iα-1aik,α-1bik=αmα-1α⋃k⋃iα-1aik,α-1bik=αm⋃k⋃iα-1aik,α-1bik=α∑km⋃iαα-1aik,α-1bik=∑kαm⋃iαα-1aik,α-1bik=∑kmN⋃i(.aik,bik).=∑kmN⋃iδik=∑kmNGk.This completes the proof.Lemma 7.
Let[.a,b]. be a closed interval in RN. If this interval is covered by a finite family Λ of open intervals (.λ,μ). in RN, namely, [.a,b].⊂N∪Λ(.λ,μ)., then b-.a<.N∑Λ‍mN(.λ,μ)..Proof.
If[.a,b].⊂∪Λ(.λ,μ)., then αα-1a,α-1b⊂α∪Λ1α-1λ,α-1μ, where Λ1 is the family of open intervals α-1(.λ,μ).=α-1λ,α-1μ in R. Thus (13)α-1a,α-1b⊂⋃Λ1α-1λ,α-1μ,and, according to this, α-1a,α-1b is covered by a finite family Λ1 of open intervals α-1λ,α-1μ in R. Then the following inequality holds:(14)α-1b-α-1a<∑Λ1mα-1λ,α-1μ=∑Λ1α-1μ-α-1λ.Therefore we have(15)αα-1b-α-1a<α∑Λ1α-1μ-α-1λand hence (16)b-.a<.α∑Λ1α-1αα-1μ-α-1λ=N∑Λαα-1μ-α-1λ=N∑Λαmα-1λ,α-1μ=N∑ΛmN(.λ,μ)..Lemma 8.
If an open intervalΔ in RN is the composition of at most numerable family of open sets in RN, namely, Δ=∪kGk, then (17)mNΔ≤∑kmNGk.Proof.
If we sayΔ=(.A,B)., then mNΔ=αα-1B-α-1A. By the proposition of real Lebesgue measure, we can write (18)αmα-1A,α-1B=αα-1B-α-1A≤α∑kmα-1Gk=α∑kα-1αmα-1Gk=N∑kαmα-1Gk=N∑kmNGk.This completes the proof.Theorem 9.
If a bounded open setG is the composition of at most numerable family of open sets in RN, namely, G=∪kGk, then(19)mNG≤N∑kmNGk.Proof.
We can easily see that(20)mNG=αmα-1G=αmα-1⋃kGk=αm⋃kα-1Gk≤α∑kα-1αmα-1Gk=N∑kαmα-1Gk=N∑kmNGk.This completes the proof.



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