A Riesz Representation Theorem for the Space of Henstock Integrable Vector-Valued Functions

Abstract

Using a bounded bilinear operator, we define the Henstock-Stieltjes integral for vector-valued functions; we prove some integration by parts theorems for Henstock integral and a Riesz-type theorem which provides an alternative proof of the representation theorem for real functions proved by Alexiewicz.

1. Introduction

Henstock in [1] defines a Riemann type integral which is equivalent to Denjoy integral and more general than the Lebesgue integral, called the Henstock integral. Cao in [2] extends the Henstock integral for vector-valued functions and provides some basic properties such as the Saks-Henstock Lemma.

Schwabik in [3] considers a bilinear form, defines a Stieltjes type integral, and performs a study about it including [4]; following his ideas we give integration by parts theorem involving a bilinear operator and, through it, we prove a representation theorem for the space of Henstock vector-valued functions.

This paper is divided into five sections; in a first step, in Section 2 we present some preliminaries and introduce the Henstock-Stieltjes integral via a bilinear bounded operator and the Bochner integral, together with some basic properties. In Section 3 we provide two useful kinds of integration by parts theorems, one of them in terms of the Bochner integral and the other using Henstock-Stieltjes integral; the representation theorem is proved in Section 4 which, if we consider real-valued functions, provides an alternative proof of the representation theorem proved by Alexiewicz (Theorem 1 in [5]).

2. Preliminaries

Throughout this paper , , and will denote three Banach spaces, , , and , which will denote their respective norms, the dual of , a bounded bilinear operator fixed, and a closed finite interval of the real line with the usual topology and the Lebesgue measure, which we denote by . For a function we denote the Lebesgue integral of on a measurable , when it exists, by .

Definition 1. is a bounded bilinear operator if is linear in each variable and there exists such that ; in this case, the norm of the operator is .

We say that is a tagged partition of if is a finite collection of nonoverlapping closed intervals whose union is such that for every . Given a function from to , called gauge on , we say that a tagged partition is -fine if

Definition 2. A function is Kurzweil integrable in if there exists such that for every there exists a gauge on such that if is a -fine tagged partition of , then We write .

Definition 3. A function is Henstock integrable in if there exists such that for every there exists a gauge on such that if is a -fine tagged partition of , then We write .

The Henstock integral is also known as Henstock-Lebesgue integral, briefly HL integral ([6]), or variational Henstock integral ([7]).

In [8] we can find some properties of both integrals such as the linearity, integrability over subintervals, and the continuity of the function , called primitive, given by or , .

Definition 4. Let and let be a subset of . (1) is said to be of strongly bounded variation (BV) on if the number is finite, where the supremum is taken over all finite sequences of nonoverlapping intervals that have endpoints in .(2) is on if is finite, where the supremum is taken over all finite sequences of nonoverlapping intervals that have endpoints in , and is the oscillation of on .(3) is said to be strongly absolutely continuous on or if for every there exists such that, for every finite or infinite sequence of nonoverlapping intervals , with , we have where for all .(4) is if for every there exists such that, for every finite or infinite sequence or nonoverlapping intervals satisfying , where for all , we have (5) is on if is the union of a sequence of closed sets such that, on each , is .

The next result gives us a characterization of Henstock integrability.

Theorem 5 (see [8, Thm. 7.4.5, pp. 217]). The function is Henstock integrable on with the primitive if and only if is continuous and on such that almost everywhere (a.e.) in , where the derivative is in the sense of Frechet.

If a function is or on , then it is bounded on ; that is, exists such that , for every . As an function is on and an function is on (immediately from the definitions), then every or function is also bounded in . It is easy to see that if is and , then is , similarly if is .

The definition of a function of strongly bounded variation can be extended considering the bilinear operator .

Definition 6. Let be a function and a partition of ; we definewhere the supremum is taken over all possible elections of , , with . where the supremum is taken over all partitions of the interval and is the strong -variation of on . If we consider in the equality (4), then we define as the -variation of on .

If or we say that is of strongly bounded -variation or is of bounded -variation, respectively.

It is straightforward that each function of strongly bounded variation is of strongly bounded -variation. We recommend the reader interested in this topic to consult the study exposed in [9].

2.1. Stieltjes-Type Integrals

As we mentioned in the introduction, Schwabik in [3] gives the next definition and proves some basic properties such as the Uniform Convergence Theorem.

Definition 7. is the Kurzweil-Stieltjes integral of with respect to if for every there exists a gauge on such that for every -fine tagged partition of .
In this case we write .

Now, we introduce the following integral.

Definition 8. A function is Henstock-Stieltjes integrable in with respect to if there exists such that for every there exists a gauge of such that if is a -fine tagged partition of , then We write .

It is immediate that every Henstock integrable function is Kurzweil integrable and its integrals are the same; we can repeat the proof of this fact for the previous Stieltjes integrals. Similarly, we can prove the properties of linearity and integrability over subintervals for the Henstock-Stieltjes integral directly of the proofs in [8] with slight changes. We omit the formulations and the proofs of such results.

Theorem 9 (see [3, Thm. 11]). Assume that the functions , , and are given. If , the Kurzweil-Stieltjes integrals exist and the sequence converges on uniformly to , then the integral exists and

2.2. Bochner Integral

Let us recall that a function is called simple if there is a finite sequence of Lebesgue measurable sets such that for and , where for , , and in this case the Bochner integral of is .

A function is strongly measurable if there exists a sequence of simple functions that converges pointwise to a.e. on .

A function is Bochner integrable if there is a sequence of simple functions , , such that a.e. in and the Bochner integral of is denoted by and is defined by

We will use the following well-known results of the Bochner integral.

Theorem 10 (see [8, Cor. 1.4.4, pp. 26]). A strongly measurable function is Bochner integrable on if there exists a function , which is Lebesgue integrable such that , .

Theorem 11 (see [8, Thm. 7.4.5, pp. 222]). A function is Bochner integrable on if and only if there exists a function , which is on such that a.e. on .

Given and we define the function , given by ; we will use this function from now on.

Lemma 12. If is a continuous function and is Bochner integrable, then the function is Bochner integrable.

Proof. Since is continuous, then it is strongly measurable; moreover there exists such that for all . Because is strongly measurable and is continuous, is strongly measurable.
By Theorem 10 the real function is Lebesgue integrable.
Now hence is Bochner integrable as a consequence of Theorem 10.

It is known that every vector-valued function which is strongly measurable is weakly measurable; that is, is measurable for each ; the inverse, in general, is not true (see [10, Example 5, Chapter II, 1, and pp. 43]) however under certain conditions is equivalent.

Theorem 13 (see [10, Thm. 2, Chapter II, 1, pp. 42] (Pettis)). Let be a function. The following conditions are equivalent: (i) is strongly measurable.(ii) is weakly measurable and there exists a measurable set with such that is separable.

Theorem 14. If is continuous a.e., then is strongly measurable.

Proof. As is continuous a.e., then is weakly continuous; that is, is continuous a.e. and, hence, measurable.
We define . is Lebesgue measurable with , as is separable and then is separable and, furthermore, is continuous, and then is separable; hence is strongly measurable by the Pettis Theorem.

Lemma 15 (see [11, Lemma 6]). If is of strongly bounded variation on , then is Bochner integrable on .

As a consequence of Lemmas 12 and 15, we have the following result.

Corollary 16. Let be Henstock integrable on and its primitive, of strongly bounded variation, then is Bochner integrable on .

It is easy to prove that the set of functions , , , and on form vector spaces with the sum and product by scalars; moreover, these spaces of functions are algebras under the bilinear operator .

Lemma 17. Let and be functions and a subset of . If is and is , then is .

Proof. Since is then , where is .
For every there exists such that, for every , with , we have and , andand then

Lemma 18. Let and be functions and a subset of . If is and is , then is .

Proof. The proof is analogous to Lemma 17 changing the inequality 2 by

3. Integration by Parts Theorem

3.1. Involving Bochner Integral

Theorem 19. Let be a Henstock integrable function with primitive , of strongly bounded variation, and the Bochner primitive of . Then exists and

Proof. By Corollary 16, is Bochner integrable. Let be a function given by is continuous due to the continuity on of , , and .
Theorem 5 implies that is on and Theorem 11 implies that is on ; hence the function is on by Lemma 18. Finally, Lemma 17 implies that is on .
In order to prove that is differentiable on , using the fact that for every , , we calculate which tends to 0 when ; hence is differentiable on and . Then By Theorem 5, exists and (15) is fulfilled.

3.2. Involving Henstock-Stieltjes Integral

Theorem 20. Let and be functions. If the integral exists and , then

Proof. Let . There exists a gauge on such that for every , which is -fine. Then,

Now, we shall prove the following Theorem, which is a consequence of Theorem 9.

Theorem 21 (uniform convergence theorem). Let , , and . If , the integrals exist for each , and the sequence converges uniformly to in , then the integral exists and

Proof. Let ; because converges uniformly to , there exists such that for every and Hence, for every ,
Theorem 9 implies the existence of the integral for every and Hence, there exists such that , for every .
Let be fixed; as the integral exists, there exists a gauge on such that, for every -fine tagged partition of , We have Hence, exists and

Theorem 22. If is a step function, then for every , the integral exists.

Proof. Analogous to the proof of [12, Lemma 3.2], is enough to prove for functions of the forms , , , and , where and . Let , , and . Given we define if and if ; then for any -fine tagged partition of , is the tag of one subinterval, if , , and ; otherwise Hence The proofs of the cases , , and are analogous.

Schwabik in [3] introduces the concept of vector-valued regulated functions; we shall only use the following characterization.

Theorem 23 (see [3, Prop. 2]). is regulated if and only if it is the uniform limit of step functions.

Theorem 24. If is regulated and with , then the integral exists.

Proof. In as much as is regulated, there exists a sequence , , of step functions which converges uniformly to , by Theorem 22; the integrals exist for each . The Uniform Convergence Theorem implies the existence of the integral .

Theorem 25 (integration by parts theorem). If is Henstock integrable, its primitive, and with ; then exists and

Proof. Let . Since is Henstock integrable, with its primitive, there exists a gauge such that if is a -fine tagged partition of , On the other hand, since is continuous, it is regulated, and by Theorem 24, exists; then there exists a gauge such that if is a -fine tagged partition of , We define a gauge by Let be a -fine tagged partition of and by the left-right process (see [13, Section 1, pp. 6]) we can assume that the tags are the left endpoint of each subinterval; then

As we can see, we have two types of integration by parts theorems, one is of the Stieltjes type and the other is non-Stieltjes; it is possible to ask for the conditions so that the integral of the Stieltjes type becomes a non-Stieltjes; for that, we must do the following analysis:(1)The essence in the proof of Theorem 19 is the derivative of the primitive of the function ,that is, the Fundamental Theorem of Calculus.(2)In Theorem 25 the Fundamental Theorem of Calculus does not apply because is not necessarily differentiable, and if it is, the primitive of , in general, is not .(3)The condition of differentiability on a function of strongly bounded variation is equivalent to and has the Radon-Nikodým property (see Distel and Uhl [10, Chapter VII 6]). Is it fulfilled with functions of strongly bounded -variation?

Therefore, the condition of is one that ensures the Fundamental Theorem of Calculus, so we have the following theorems.

Theorem 26. Let be continuous function and with such that exists on . Then

Proof. exists by Theorem 24; then given , we have a gauge and a -fine tagged partition of . Since exists, for every there exists such that if then We define a gauge by . For all which is -fine tagged partition of and supposing that each tag is the left endpoint of its respective subinterval we have where is the bound of due to it is continuous.

Obviously, we can change the condition over in Theorem 26 if we ask for strongly bounded variation and impose the Radon-Nikodým property on ; either with these conditions or with those of Theorem 26 we can write equality (28) as

A function satisfies the Lipschitz condition if there exists such that , , , and is of bounded slope variation (BSV) if is bounded for all divisions .

Lee in [14, p. 75] proves that a function is the primitive (in the sense of Henstock) of a function of strongly bounded variation on if and only if satisfies the Lipschitz condition and is of bounded slope variation on ; this same characterization can be extended for our case (see the proof of [15, Thm. 10]). So we can see that the Fundamental Theorem of Calculus applies and we can restate the above integration by parts formula as follows.

Corollary 27. Let be a function and its primitive. If satisfies the Lipschitz condition and is of bounded slope variation, then exists and

4. Representation Theorem

Now, we will establish an important connection between the space of Henstock integrable functions and its dual space: a Riesz representation theorem.

Definition 28. Let be the space of all Henstock integrable functions from to . We define a norm on , called Alexiewicz norm (see e.g., [5] or [16]), by

Theorem 29. Let be a function. If , then defines a continuous linear operator on the space of Henstock integrable functions into with

Proof. Let the primitive of and ; by Theorem 24 there exists a gauge such that if is -fine, By the integration by parts theorem we have It follows that So is bounded and continuous.

denotes the space of continuous linear operators from to and is the bilinear bounded operator given by .

Theorem 30. Let be a linear continuous operator defined on the space of the Henstock integrable functions. There exists a function with such that for every .

Proof. For each and we have . We define a function by In this form, is linear and it is also continuous due to where .
We define a function by Now, , indeed, for every arbitrary partition and for every , , with , Suppose that is a step function; by Theorem 22 we have that the integral exists.
Given , we take its primitive . Since is continuous, there exists sequence of linear piecewise functions such that uniformly on . The integral exists for every by Theorem 24.
By the Uniform Convergence Theorem, the integral exists; hence Let be the sequence of derivatives of ; is simple for each , since uniformly by the continuity of and the Integration by parts Theorem 25 we have that

Finally, we have the following representation theorem.

Corollary 31. is a linear continuous operator if and only if there exists a function with such that for every .

Consider a Banach space and in the definition of strongly bounded -variation function; then this is equivalent to the definition of strongly bounded variation function, so we have the next result.

Corollary 32. is a linear continuous functional on if and only if there exists a function and such that for every .

Hence, the dual space of is isometrically isomorphic to the space of functions of strongly bounded variation.

Since the space of Henstock integrable real-valued functions coincides with the Kurzweil integrable functions we will name the integral as the integral of Henstock-Kurzweil.

As a particular case of Corollary 31 (with ) we have obtained the following representation theorem for the space of Henstock-Kurzweil integrable functions.

Corollary 33. is a linear continuous functional if and only if exists a function , , such that for every Henstock-Kurzweil integrable function.

Remarks 34. The result above was proved in [5] by Alexiewicz; later other proofs of the theorem arose, for example, those provided by Sargent and Lee (Theorem 4 in [17] and Theorem 12.7 in [14, pp. 76], resp.), who use different techniques from those used in this work; for example, Hahn-Banach Theorem is not necessary for the proof of Theorem 30.

The Integration by parts Corollary 27 yields a new representation theorem without using Stieltjes integral, which we shall establish next. The proof of the first result below is analogous to Theorem 29; using equality (35), we will only sketch the proof of the second result.

Theorem 35. If is of strongly bounded -variation and differentiable and is bounded, then is a continuous linear operator on the space into .

If is of bounded slope variation and satisfies the Lipschitz condition then the previous theorem is also true; we shall prove the second part in the new representation theorem.

Theorem 36. Let be a linear continuous operator defined on the space of the Henstock integrable functions. There exists a function of bounded slope variation and Lipschitz such that , for every .

Proof. Let and be the primitive of . We define the function ; this function is in , and we define by ; is of bounded slope variation and satisfies the Lipschitz condition (see the proof of [15, Thm. 10]); hence it is differentiable and is of strongly bounded variation. Following the proof of Theorem 30 using which is also of strongly bounded variation, by Theorem 26 the integral is equal to and by integration by parts Corollary 27, we have

The last representation theorem identifies with the space of primitives of the functions of strongly bounded variation, unlike the Corollary 31 which identifies it with the space of functions of strongly bounded -variation.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research has been able to see the light thanks to the help, comments, suggestions, and unconditional support of Professor Lee Peng Yee and the work team and the staff of MME/NIE of Nanyang Technological University in Singapore. The authors will be eternally grateful. This research has been supported by Conacyt, VIEP-BUAP, DGRIIA-BUAP, and the Academic Group of Mathematical Modeling and Differential Equations-FCFM-BUAP.