Non-Newtonian Comment of Lebesgue Measure in Real Numbers

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.

1. Introduction

A generator is a one-to-one function , whose domain is , the set of all real numbers, and whose range is a subset of . Identity function and exponential function can be given as examples of generators. We denote by , called non-Newtonian real line, the range of generator . Non-Newtonian arithmetic operations on are represented as follows [1–5]: -addition   -subtraction   -multiplication   -division   -order  

A closed interval on is represented by and similarly an open interval can be represented [2, 6]. Let be a subset in . A point is called an interior point of if there exists an interval with . A set is said to be open in if all of its points are interior points. The component interval of an open set in is an open interval providing conditions , , and . It is known that any open set in is written as any composition of the component intervals, and also it would be an easy detection that the same thing is valid in [6–8].

In this work, the symbol in is used instead of the usual summation symbol .

This work is deviated mainly to establish a measure in Lebesgue’s sense on .

The measure of an open interval in is the length 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 measure of an open interval in is defined by One can restate the above relation as follows:

Lemma 2. If a finite number of pairwise disjoint open intervals , , are contained in the open interval , then

Proof. We may suppose that . Then, by the properties of the generator and the definition non-Newtonian sum, we haveThis completes the proof.

Corollary 3. If an infinite number of pairwise disjoint open intervals , , in , are contained in the open interval , then

Definition 4. The measure of a nonvoid bounded open set in is the sum of the measures of all its component intervals :

Here it should be noted that where . Also it should be noted thatsince the generator is a one-to-one function, and here the intervals are the component intervals of .

Theorem 5. Let and be two bounded open sets in . If , then .

Proof. We can write and as a composition of the component intervals and . Since each is in one and only one , we have This completes the proof.

Theorem 6. If the bounded open set is the composition of at most numerable family of pairwise disjoint open sets , then .

Proof. If each is written as a composition of the component intervals , thenThis completes the proof.

Lemma 7. Let be a closed interval in . If this interval is covered by a finite family of open intervals in , namely, , then .

Proof. If , then , where is the family of open intervals in . Thus and, according to this, is covered by a finite family of open intervals in . Then the following inequality holds:Therefore we haveand hence

Lemma 8. If an open interval in is the composition of at most numerable family of open sets in , namely, , then

Proof. If we say , then By the proposition of real Lebesgue measure, we can write This completes the proof.

Theorem 9. If a bounded open set is the composition of at most numerable family of open sets in , namely, , then

Proof. We can easily see that This completes the proof.

Competing Interests

The authors declare that they have no competing interests.