American Journal of Modern Physics
Volume 4, Issue 5-1, October 2015, Pages: 24-34

Measurable Iso-Functions

Svetlin G. Georgiev

Department of Mathematics, Sorbonne University, Paris, France

Svetlin G. Georgiev. Measurable Iso-Functions. American Journal of Modern Physics. Special Issue: Issue I: Foundations of Hadronic Mathematics. Vol. 4, No. 5-1, 2015, pp. 24-34. doi: 10.11648/j.ajmp.s.2015040501.13

Abstract: In this article are given definitions definition for measurable is-functions of the first, second, third, fourth and fifth kind. They are given examples when the original function is not measurable and the corresponding iso-function is measurable and the inverse. They are given conditions for the isotopic element under which the corresponding is-functions are measurable. It is introduced a definition for equivalent iso-functions. They are given examples when the iso-functions are equivalent and the corresponding real functions are not equivalent. They are deducted some criterions for measurability of the iso-functions of the first, second, third, fourth and fifth kind. They are investigated for measurability the addition, multiplication of two iso-functions, multiplication of iso-function with an iso-number and the powers of measurable iso-functions. They are given definitions for step iso-functions, iso-step iso-functions, characteristic iso-functions, iso-characteristic iso-functions. It is investigate for measurability the limit function of sequence of measurable iso-functions. As application they are formulated the iso-Lebesgue’s theorems for iso-functions of the first, second, third, fourth and fifth kind. These iso-Lebesgue’s theorems give some information for the structure of the iso-functions of the first, second, third, fourth and fifth kind.

Keywords: Measurable Iso-Sets, Measurable Is-Functions, Is-Lebesgue Theorems

1. Introduction

Genious idea is the Santilli’s generalization of the basic unit of quantum mechanics into an integro-differential operator  which is as positive-definite as +1 and it depends of local variables and it is assumed to be the inverse of the isotopic element

and it is called Santilli isounit. Santilli introduced a generalization called lifting of the conventional associative product  into the form

Called isoproduct for which

For every element  of the field of real numbers, complex numbers and quaternions. The Santilli isonumbers are defi ned as follows: for given real number or complex number or quaternion ,

With isoproduct

.

If , the corresponding isoelement of  will be denoted withor .

With  we will denote the field of the is-numbers  for which  and basic isounit .

In [1], [3]-[12] are defined isocontinuous isofunctions and isoderivative of isofunction and in [1] are proved some of their properties. If  is an isoset in , the class of isofunctions is denoted by  and the class of isodifferentiable isofunctions is denoted by , with the same basic isounit , it is supposed

.

Here  is the corresponding real set of . If is an independent variable, then the corresponding lift is , if  is real-valued function on , then the corresponding lift of first kind is defined as follows

and we will denote it by .

In accordance with [1], the isodifferential is defined as follows

.

Then

,

.

In accordance with [1], the first is-derivative of the is-function  is defined as follows

When , then

Our aim in this article is to be investigated some aspects of theory of measurable iso-functions. The paper is organized as follows. In Section 2 are defined measurable iso-functions and they are deducted some of their properties. In Section 3 is investigated the structure of the measurable iso-functions.

2. The Definition and the Simplest Properties of Measurable Is-Functions

We suppose that  is a given point set, ,  for every   be a given constant,  bea given real-valued function. With  we will denote the corresponding is-function of the first, second, third, fourth and fifth kind. More precisely,

1.   , when  is an is-function of the first kind.

2.   , when  for  when  is an is-function of the second kind.

3.   ,  for  when  is an is-function of the third kind.

4.    when  for , when is an is-function of the fourth kind.

5.   ,  for  when  is an is-function of the fifth kind.

For  with  we will denote the set

We define the symbols , , ,  and etc., in the same way.

If the set on which the is-function  is defined is designated by a letter C or D, we shall write >a) or >a).

Definition 2.1. The is-function  is said to be measurable if

1.   The set A is measurable.

2.   The set  is measurable for all .

Theorem 2.3. Let be a measurable is-function defined on the set A. If B is a measurable subset of A, then the is-function , considered only for, is measurable.

Proof. Let  be arbitrarily chosen and fixed. We will prove that

(1)

Really, let  be arbitrarily chosen. Then  and (x)>a. Since , we have that . From  and  it follows that . Because  was arbitrarily chosen and for it we get that it is an element of the set , we conclude that

(2)

Let now  be arbitrarily chosen. Then  and . Hence  and . Therefore  Because  was arbitrarily chosen and we get that it is an element of we conclude that

.

From the last relation and from (2) we prove the relation (1).

Since the iso-function  is a measurable function on the set A, we have that  is a measurable set. As the intersection of two measurable sets is a measurable set, we have that  is a measurable set. Consequently, using (1), the set >a) is measurable set. In this way we have

1.   B is a measurable set,

2.    is a measurable set for all .

Therefore the iso-function  , considered only for , is a measurable is-function.

Theorem 2.4. Let  be defined on the set A, which is the union of a finite or denumerable number of measurable sets , . If  is measurable on each of the sets , then it is also measurable on A.

Proof. Let  be arbitrarily chosen. We will prove that

(3)

Let  be arbitrarily chosen. Then  and . Since  and , there exists k such that . Therefore  and . Hence, and . Because  was arbitrarily chosen and for it we get that it is an element of  , we conclude that

(4)

Let now  be arbitrarily chosen. Then there exists l such that >a). From here  and  Hence,  and  Consequently  . Because  was arbitrarily chosen and for it we get that it is an element of  we conclude that

From the last relation and from (4) we prove the relation (3).

Since the union of finite or denumerable number of measurable sets is a measurable set, using that the sets  are measurable, we obtain that A and  are measurable sets. Therefore  is a measurable is-function.

Definition 2.5. Two is-functions  and , defined on the same setr A, are said to be equivalent if

=0.

We will write

.

Remark 2.6. There is a possibility  and in the same time.

Let

Then

.

On the oth-1+er hand,

We have that

Or

Remark 2.7. There is a possibility  and in the same time . Let

Then

.

On the other hand,

=.

Then

Therefore

Hence,

Consequently

.

Proposition 2.8. The fuAnctions f and g are equivalent if and only if the functions  and  are equivalent

Proof. We have

Definition 2.9. Let some property P holds for all the points of the set A, except for the points of a subset B of the set A. If  , we say that the property P holds almost everywhere on the set A, or for almost all points of A.

Definition 2.10. We say that two is-functions defined on the set A are equivalent if they are equal almost everywhere on the set A.

Theorem 2.11. If  is a measurable is-function defined on the set A, and if , then the is-function  is also measurable.

Proof. Let

Because  we have that

or

Since every function, definite on a set with measure zero is measurable on it, we have that the is-function  is measurable on the set B.

We note that the is-functions  and  are identical on D and since the is0-function  is measurable on D, we get that the is-function  is measurable on D.

Consequently the is-function is measurable on

Theorem 2.12. If the is-function , defined on the set A, is measurable, then the sets

),

Are measurable for all .

Proof. We will prove that

(5)

Really, let  be arbitrarily chosen. Then  and . Hence, for every  we have . Therefore

.

Because  was arbitrarily chosen and for it we obtain

We conclude that

(6)

Let now  be arbitrarily chosen. Then  for every natural number n. From here  and

For all natural number n. Consequently

or

and . Since  was arbitrarily chosen and we get that , we conclude

From the last relation and from (6) we obtain the relation (5).

Because the intersection of denumerable measurable sets is a measurable set, using the relation (5) and the fact that all sets  are measurable for all natural numbers n, we conclude that the set  is a measurable set.

The set  is a measurable set because

The set  is measurable set since

The set  is measurable since

Remark 2.13. We note that if at least one of the sets

),

Is measurable for all , then the iso-function  is measurable on the set A.

Really, let  is measurable for all . Then, using the relation

(7)

we obtain that the set  is measurable for all .

If  is measurable for all , then using the relation

we get that the set  is measurable for all .

If  is measurable for all , then using the relation

,

We conclude that the set  is measurable for all .

Theorem 2.14. If  for all points of a measurable set A, then the is-function  is measurable.

Proof. For all  we have that

.

Since the sets A and  are measurable sets, then  is measurable for all . Therefore the is-function  is measurable.

Definition 2.15. An is-function  defined on the closed interval [a, b] is said to be a step is-function if there is a finite number of points

Such that  is a constant on , .

Proposition 2.16. A step is-function is measurable.

Proof. Let (x) is a step is-function on the closed interval [a, b]. Let also,

be such that  is a constant on , . From the previous theorem we have that  is measurable on , . We note that

the sets , , are sets with measure zero. Therefore the is-function

is measurable on , . From here, using that

We conclude that the is-function  is measurable on [a, b].

Theorem 2.17. If the is-function , defined on the set A is measurable and  then the is-functions

1.   ,

2.

3.

4.

5.

are also measurable.

Proof. Let  be arbitrarily chosen. The assertion follows from the following relations.

1.

2.    if c>0,  if c<0.

3.    if a<0,  if

4.    if a<0,  if .

5.    if a>0,  if a<0,  if a=0.

Definition 2.18. An is-function , defined on the closed interval [pa, b], is said to be is-step is-function, if there is a finite number of points

Theorem 2.19. Let >0 for every  and  is measurable on [a, b]. Let also,  is an iso-step is-function on [a, b]. Then  is measurable on [a, b].

Proof. Let

be

From the last theorem it follows that  is a measurable is-function on ,  Fromn-1 here and from

Since {b} is a set with measure zero, we conclude that the is-step is-function  is measurable on [a, b].

Definition 2.20. Let M be a subset of the closed interval [a, b]. The function  for  and  for  is called the characteristic function of the set M.

Theorem 2.21. If the set M is a measurable subset of the closed interval A=[a, b], then the characteristic function  is measurable on [a, b].

Proof. The assertion follows from the following relations.  if   if  if a<0.

Definition 2.22. Let M be a subset of the set A=[a, b]. The iso- function  if  and  if , will be called characteristic is-function of the set M.

Theorem 2.23. Let  be a measurable function on A=[a, b], M be a measurable subset of A. Then the characteristic is-function  of the set M is measurable.

Proof. Let  be arbitrarily chosen. Then

,

From here, using that the sets  and  are measurable sets, we conclude that  is a measurable set. Because the constant a was arbitrarily chosen, we have that the characteristic function  is a measurable is-function.

Theorem 2.24. Let f and  are continuous functions on the closed set A. Then the is-function  is measurable.

Proof. Let  be arbitrarily chosen. Since every closed set is a measurable set, we conclude that the set A is a measurable set.

We will prove that the set  is a closed set.

Let  be a sequence of elements of the set  such that

Since  is a subset of the set A we have that . Because the set A is a closed set, we obtain that . From the definition of the set  we have that

Hence, when , using that f and  are continuous functions on the set A, we get

i.e., . Therefore the set is a closed set. From here, the set  is a measurable set. Because the difference of two measurable sets is a measurable set, we have that the set

Is a measurable set.

Since  was arbitrarily chosen, we obtain that the is-function of the first kind  is measurable.

Theorem 2.25. Let f and  are continuous functions on the closed set A. The the is-functions

are measurable on A.

Theorem 2.26. If two measurable is-functions  and  are defined on the set A, then the set  is measurable.

Proof. We enumerate all rational numbers

.

We will prove that

(8)

Let

Be arbitrarily chosen. Then

There exists a rational number  such that

Therefore

i.e.,

).

Consequently

)

And

Because  was arbitrarily chosen and for it we get
we conclude that

(9)

Let no

be arbitrarily chosen. Then there exists a natural k so that

.

Hence,

Then

or

Therefore

Because

Was arbitrarily chosen and for it we get that , we conclude that

From the last relation and from the relation (9) we get the relation (8).

Since  and  are measurable iso-functions on A, we have that the sets

are measurable sets for every natural k, whereupon the sets

Are measurable sets for every natural k.

Therefore, using the relation (8), we obtain that the set  is a a measurable set.

Theorem 2.27. Let  and  be finite measurable is-functions on the set A. Then each of the is-functions

1.

2.

3.

4.    if  on A,

Is measurable.

Proof.

1.         Let  be arbitrarily chosen. Since  is measurable, then  is measurable. From here and from the last theorem it follows that the set

Is measurable. Because was arbitrarily chosen, we conclude that the function is measurable.

2.         Since  is a measurable is-function, we have that the function  is a measurable is-function. From here and from 1) we conclude that the is-function

Is measurable.

3.         We note that

(10)

Since  and  are measurable iso-functions, using 1) and 2) we have that

Are measurable is-functions. Hence the is-functions

Are measurable, whereupon

Are measurable. From here, using 1) and (10), we conclude that  is measurable.

4.         Since  is measurable and  on A, we have that the is-function  is measurable. From here and from 3) the is-function

Is measurable.

Theorem 2.28. Let  be a sequence of measurable is-functions defined on the set A. If

(11)

Exists for every then the is-function  is measurable.

Proof. Let  be arbitrarily chosen. For   we define the sets

We will prove that

(12)

Let

Be arbitrarily chosen. Then

Hence, there is enough large natural number  such that

Using (11), there are enough large natural numbers k and m such that

i.e.,

From here, it follows that there is enough large n so that  for every , i.e.,  and then

Since  was arbitrarily chosen and we get that it is an element of the set , we conclude that

(13)

Let now  be arbitrarily chosen.

Then, there are  so that

or

.

Hence,

or

Therefore

Since was arbitrarily chosen and for it we obtain , we conclude that

From the last relation and from (13) it follows the relation (12).

Since  are measurable, we have that the sets  are measurable for every  hence  are measurable for every  and then, using (12), the set  is measurable. Consequently the is-function  is measurable.

Theorem 2.29. be a sequence of measurable is-functions defined on the set A. If

(14)

Exists for almost everywhere then the is-function  is measurable.

Proof. Let B be the subset of A so that the relation (14) holds for every . From the previous theorem it follows that the is-function  is measurable on the set B.

We note that

Therefore the is-function  is measurable on . Hence, the is-function  is measurable on A.

Let

Then

1.

2.

If

3.

If

4.

If

5.

If

.

3. The Structure of the Measurable Is-Functions

Theorem 3.1. (is-Lebesgue theorem for is-functions of the first kind) Let there be given a sequence  of measurable functions on a set A, all of which are finite almost everywhere. Let also,  be a sequence of measurable functions on the set A,

For all natural numbers n and for all  where  and  are positive constants. Suppose that

Almost everywhere on the set A, and  is finite almost everywhere on A,

For all  Then

For all

Proof. We will note that the limit functions f(x) and  are measurable and the sets under considerations are measurable.

Let

Since

,

using the properties of the measurable sets, we have that

Let

We have that

.

Hence,

Let us assume that Then, using the definition of the set  , we have

Since

we have that

and their limit

are finite.

Therefore there is an enough large natural n such that

for every . Then where  and from here

Consequently

Because , from the last relation, we have that  i.e.,

,

and since

,

or

.

As in above one can prove the following results for the other kinds of is-functions.

Theorem 3.2. (is-Lebesgue theorem for is-functions of the second kind) Let there be given a sequence  of measurable functions on a set A, all of which are finite almost everywhere. Let also,  be a sequence of measurable functions on the set A,

For all natural numbers n and for all  where  and  are positive constants. Suppose that

Almost everywhere on the set A, and  is finite almost everywhere on A,

For all  Then

for all

Theorem 3.3. (is-Lebesgue theorem for is-functions of the third kind) Let there be given a sequence  of measurable functions on a set A, all of which are finite almost everywhere. Let also,  be a sequence of measurable functions on the set A,

For all natural numbers n and for all  where  and  are positive constants. Suppose that

Almost everywhere on the set A, and  is finite almost everywhere on A,

For all  Then

for all

Theorem 3.4. (is-Lebesgue theorem for is-functions of the fourth kind) Let there be given a sequence  of measurable functions on a set A, all of which are finite almost everywhere. Let also,  be a sequence of measurable functions on the set A,

For all natural numbers n and for all  where  and  are positive constants. Suppose that

Almost everywhere on the set A, and  is finite almost everywhere on A,

For all  Then

For all

Theorem 3.5. (is-Lebesgue theorem for is-functions of the fifth kind) Let there be given a sequence  of measurable functions on a set A, all of which are finite almost everywhere. Let also,  be a sequence of measurable functions on the set A,

For all natural numbers n and for all  where  and  are positive constants. Suppose that

Almost everywhere on the set A, and  is finite almost everywhere on A,

For all  Then

for all

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