#### Abstract

The aim of this paper is to extend the notion of -Riemann integrability of functions defined over to functions defined over a rectangular box of . As a generalization of step functions, we introduce a notion of -step functions which allows us to give an equivalent definition of the -Riemann integrable functions.

#### 1. Introduction

The motivation of this work results from the notion of the -Riemann integral of a function defined over an interval of recently introduced by Olbrys’ [1]. Our aim of this paper is to give an equivalent definition of the -Riemann integrability by the use of -step functions and extend this notion to functions defined over a rectangular box of .After introducing the notion of -multiple integral, we give its properties and we show a -version of Fubini’s theorem.

Throughout this paper, denotes a subfield of the field of real numbers . We denote the set of positive elements of by . For two real numbers , we denote If is a rectangular box of , we denote Following [1], we define the set of the -partitions of the interval by As in [1], we give a generalization of the radially -continuous and uniformly radially -continuous functions and propositions easy to prove.We say that is radially -continuous if it is radially -continuous at every point.

*Definition 1. *A function is called radially -continuous at a point if for every

*Definition 2. *We say that a function is uniformly radially -continuous if for any and , the mapping
is uniformly continuous.

Every continuous and any uniformly continuous function in the usual sense is radially -continuous and uniformly radially -continuous, respectively.

Proposition 3. (i)*A function is radially -continuous if and only if for every subrectangular box of , the function is continuous*(ii)*A function is uniformly radially -continuous if and only if for every subrectangular box of , the function is uniformly continuous*

#### 2. **-**Step Functions and -Riemann Integral

In the following definition, we introduce the notion of the -step functions which are a generalization of step functions.

*Definition 4. *A function is said to be a -step function on , if there exists a *-*partition of for which is constant on each set . In this case, we say that is a -compatible partition of on .

We denote the set of the -step functions on by .

*Remark 5. *(i)Every -step function on is bounded on .(ii)If , we get the usual definition of a step function.

*Example 1. *(i)Every constant function on is a -step function.(ii)Consider the following functions:It is easy to show that is a -step function but it is not a -step function on , and is a -step function, but it is not a -step function on .

Now, suppose that is a -step function on and let be a -compatible partition of , with whenever for all . Consider the following number:

We can easily show that the number does not depend on the choice of the -compatible partition.

*Definition 6. *Let be a -step function on , and a -compatible partition of . The real number is called the -integral of on and is denoted by

As for the introduction of the usual Riemann integral, we extend the definition of the -Riemann integral to a broader class of functions than -step functions.

*Definition 7. *A function is said to be -Riemann integrable on , if for every , there exist two -step functions on such that

We denote the set of the -Riemann integrable functions by .

*Remark 8. *One can easily verify that is a part of .

Let be a bounded function on and set

Note that and

Proposition 9. *Let be a bounded function on . Then, if and only if *

*Proof. *Suppose that . Given , then there exist such that
Since
then,
Therefore,
Conversely, let , then there are such that
Then,
Therefore, .

*Definition 10. *Let , the common value is called the *-*integral of on ; it is denoted by

In the case when , we will use the standard symbol instead of .

Now, we will show that our definition of the -integral is equivalent to that recently introduced by Olbrys’ ([1], Definition 7)

Proposition 11. *A function is -Riemann integrable on in the sense of Definition 7 if and only if it is -Riemann integrable on in the sense of ([1], Definition 7) (meaning the two definitions are equivalents). Furthermore, when , we have
where
*

*Proof. *Suppose that in the sense of ([1], Definition 7). Given , then there exists a partition of such that
where and . We have thus built two functions such that
where and are defined on by and
Therefore, is -Riemann integrable on in the sense of Definition 7.

Conversely, suppose that is -Riemann integrable on in the sense of Definition 7. Then, there exist and such that
Let be a common -compatible partition of and , and let
Then,
which shows
This means that is -Riemann integrable on in the sense of ([1], Definition 7).

Finally, if , then the equalities (19) are ensured by Proposition 9 and the inequalities (26).

#### 3. -Multiple Integral

This section is devoted to -multiple Riemann integral. For the theory of the classical multiple Riemann integral, see for instance [2].

Let be a closed rectangular box in , where are real numbers such that for all .

We denoted by

*Definition 12. *A rectangular box is said to be a -subrectangular box of if for all .

*Example 2. * is a -subrectangular box of .

is not a -subrectangular box of .

*Definition 13. *A -partition of is a finite collection of -subrectangular box of such that
where is the interior of .

*Definition 14. *(i)Let be a bounded function over . is said to be a -step function on if there exists a -partition such that is constant on for all In this case, is said to be a -compatible partition of on We denoted the set of the -step functions on by .
(ii)A -refinement of a -partition of is another -partition of such that each is a -subrectangular box of some

Let be a -step function and be a -compatible partition of such that for all . We define the number

We will first show that does not depend on the choice of the -compatible partition .

Lemma 15. *Let be a -step function and be a -compatible partition of . Then, for every -refinement of , we have
*

*Proof. *Let be a -compatible partition of such that for all and be a -refinement of . For all , there is such that Then,

Corollary 16. *Let be a -step function and be a -compatible partition of . The number does not depend on the choice of the -compatible partition .*

*Proof. *Let and be two -compatible partitions of , then , where , is a -refinement of and . Therefore, by using Lemma 15, we get

*Definition 17. *Let be a -step function and be a -compatible partition of . We called the ** -**Riemann integral of on and we denote it by

In the following proposition, we present some basic properties of -Riemann multiple integral of -step functions.

Proposition 18. *Let be two -step functions on and . Then,
*(1)*The function is a -step function on ; moreover,**(2)**If , then *(3)*The absolute value is a -step function on and *(4)*If is a -partition on , then *

*Proof. *(1) Let be a -compatible partition of , then , where , is a -compatible partition of ; moreover, we have
(2) and (3) are evident.

(4) Let be a -compatible partition of such that for all . Then,
Since is a -compatible partition of on . Then,

##### 3.1. -Riemann Integrable Functions Defined over

*Definition 19. *Let be a function, we said that is a -Riemann integrable function, if for every , there exist two -step functions and on such that

We denote the set of the -Riemann integrable functions by .

*Remark 20. *(i)Every -Riemann integrable function on is bounded on (ii) is a part of

Now, suppose that is a bounded function on . Set

Note that for all and

Theorem 21. *Let be a bounded function on . Then, if and only if .*

*Proof. *Suppose that . Given , there exists and in such that
Then,
which shows that
for all Therefore,
Conversely, suppose that . Let , by the definition of the greatest lower bound and the least upper bound, there are such that
Then,
Therefore,

*Definition 22. *Let , the common value is called the *-*Riemann integral of on , and we denote it by

In the case when , we will use the standard symbol instead of .

Proposition 23. *Let be a function. Then, is -Riemann integrable on if and only if there exist two sequences of -step functions and such that
*

Moreover, we have Before presenting a version of the Fubini theorem for -Riemann integrable functions. We start by giving a lemma that is essential for the proof of this result.

*Proof. *It runs in a similar way as for the case of one variable. Suppose that is -integrable. Then, For all , there exist two -step functions and on such that
Hence, . Now, let us show that
Let . We have
Then,
Hence,
The converse is trivial.

Lemma 24. *Let be two subfields of and let be a -step function. Then, is a -Riemann integrable function. Moreover, we have
*

*Proof. *Let be a -compatible partition of such that for all . For all , there is a sequence of -subrectangular of such that
Set
where is the closure of . Then, there is a sequence of -subrectangular of such that
Now, consider the functions
We have is a -compatible partition of and on ; moreover,
Since is a decreasing sequence, then then Hence, is -Riemann integrable. Moreover,

Theorem 25. *Let be two subfields of . Then, and
*

*Proof. *Let , given , there exist such that
By Lemma 24, there exist two sequences and of -step functions such that
with
Then, there is such that
Hence,
Therefore, and

Corollary 26. *Let be a uniformly radially -continuous function, then it is -Riemann integrable on any subrectangular of .*

*Proof. *Let be a subrectangular of . By Proposition 3, is uniformly continuous on . Since , where is the closure of , then there exists a unique continuous function such that . Since is Riemann integrable on then, using Theorem 25, is -Riemann integrable on ; moreover,

Lemma 27. *Let be a -step function on . Then, there is a -partition of and a -partition of such that the function
is a -step function and is a -compatible partition of whenever . Moreover, the function
is a -step function on and
*

*Proof. *Let be a -partition of , then there exists a -refinement of such that and are two -partitions of and , respectively. Since is a -step function on , then there are two -partitions and of and , respectively, such that is a -partition of and is constant on each ; hence, is a -step function on .Now, we can say that is a -step function on .Let us show the last equality in the theorem. Consider the two -partitions and given in the first step, then for all and , there exist such that whenever ; therefore, we have

Theorem 28 (Fubini theorem). *Let be a -Riemann integrable function on . If the function
is -Riemann integrable on for all , then the function
is a -Riemann integrable function on and
*

*Proof. *Let , there exist two -step functions and on such that
Now, we can find two -step functions and such that
Since