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 then is -Riemann integrable on ; moreover,
3.2. Biconvex Function
Now, we give the notion of -biconvex function and we will prove that any -biconvex function is -Riemann integrable.Since a function is Jensen-convex if and only if it is -convex, for more details, see [3, 4]. Then, we can easily show that a function is Jensen-biconvex if and only if it is -biconvex.
Definition 29. A mapping is said to be -bilinear, if it satisfies (i)(ii)for every and for every .
Definition 30. A function is said to be (1)Jensen-biconvex if(i) for every and (ii) for every and (2)-biconvex if(i), for every and and (ii) for every and and
Now, we claim that every -biconvex function is -Riemann integrable. For the proof, we need the following lemma (see [5] p.143)
Lemma 31. Let be an open interval and be a convex function. Then, for every and such that and .
Theorem 32. Let be a bounded and -biconvex function, and let be a closed rectangular box such that . Then, is uniformly continuous on .
Proof. Put , there exists a positive number such that with and . We show easily that It is sufficient to show that . If , then there exists such that . Put , then . Moreover, Then, On the other hand, since is bounded, there exist two real numbers such that Now, given an arbitrary , put . We have and Further, put . Take arbitrary such that Then, there exist , such that Whence and by (88), we get Consider the numbers By (92), we have Then, . Now, Since the functions are -convex, taking in Lemma 31, and , we get and taking , and , we find that Since then which mean, by using (87), that Hence, is uniformly continuous on .
Corollary 33. Let be a bounded and -biconvex function. Then, is -Riemann integrable on any subrectangular such that
Proposition 34. Let be a -bilinear function. Then, it is -Riemann integrable on every rectangular box . Moreover,
Proof. is a -bilinear function; then, it is a -biconvex function; hence, is a -Riemann integrable function. Since are -convex function, then by ([1], Theorem 7), they are -Riemann integrable functions; moreover, By Fubini Theorem 3.18, we get
Remark 35. In this last part, we only took the case of two variables. By the same way, we can extend these results to the case of variables.
4. Conclusion
We plan, thereafter, to extend the definition of -integrability to functions defined on bounded subsets and to generalize Fubini’s theorem.
Data Availability
Data are available in https://link.springer.com/article/10.1007/s00010-017-0472-0.
Conflicts of Interest
The authors declare that they have no conflicts of interest.