The space of all gauge integrable functions equipped with the Alexiewicz norm is not a Banach space. This article defines a class of gauge integrable functions that pose a complete normed space under a suitable norm. First, we discuss how the introduced space evolves naturally from the properties of the gauge integral. Then, we show that this space properly contains the space of all Lebesgue integrable functions. Finally, we prove that it is a Banach space. This result opens the gate for further studies, such as investigating the verity of convergence theorems on the presented space and studying its other properties.

1. Introduction

The gauge integral was formulated and developed separately by Ralph Henstock and Jaroslav Kurzweil in [1, 2], respectively. Therefore, the gauge integral is referred to as Henstock–Kurzweil integral in much literature. Their integration method outweighs Reimann and Lebesgue integrals in different areas and shows great potential. In the first section of this article, we will introduce some basic definitions and important results regarding the gauge integral. Throughout this article, will denote a nondegenerate finite closed interval.

Definition 1. A gauge onis a strictly positive function.

Definition 2. Letbe a tagged partition of, and letbe a gauge on. We say thatis-fine if for all, and we have

If is any gauge on , then Cousin’s lemma (see [3]) guarantees the existence of a tagged partition of , such that is -fine. This result allows us to introduce the following main definition.

Definition 3. A functionis said to be gauge integrable if there exists a real numbersuch that for all, there exists a gaugeonsatisfying the following; ifis a tagged partition ofandis-fine, thenIn this case, we write

It is clear that the gauge integral is a generalization of the Riemann Integral, where the constant in Riemann’s definition is being replaced with a positive function (gauge) . This slight yet crucial change makes it possible to integrate a much broader class of functions. In fact, every Lebesgue integrable function is a gauge integrable function, but the reverse is not valid. There exist functions that are gauge integrable, and at the same time, they are not Lebesgue integrable nor Riemann integrable.

A significant difference between the gauge integral and the Lebesgue integral appears in regard to the fundamental theorem of calculus, where the gauge integral allows a more general form of the fundamental theorem of calculus. The following theorem illustrates this difference where we combine some results (see [4]).

Theorem 1. Letbe a continuous function, which is differentiable nearly everywhere and, then(i)is gauge integrable on(ii)is Lebesgue integrable onif and only ifis absolutely continuous

Another superior feature of the gauge integral is concerning the improper integral, where a function is gauge integrable if and only if its improper gauge integral exists. This feature is well known as Hake’s theorem (see [5]), which can be stated as follows.

Theorem 2. Letbe a function, thenis gauge integrable onif and only ifis gauge integrable onfor allandexists. In this case,

Definition 4. Letbe a gauge integrable function, then

The above norm is called Alexiewicz’s norm, and it was introduced in [6]. Despite many attempts to define different norms on the class of all gauge integrable functions, Alexiewicz’s norm became the standard norm and got the most attention. It is a seminorm, but since a. e, is considered a norm once we identify the functions which are equal almost everywhere.

Definition 5.

is a normed space, but unfortunately, it is not complete; it is not a Banach space. Its completion is described as a subspace of distributions (see [3]). This lack of completeness opens many areas of research. One of these areas is the search for complete subspaces of , which strictly contain the space of all Lebesgue integrable functions.

2. Construction and Results

As it is mentioned above, every Lebesgue integrable function is a gauge integrable function. However, if is a gauge integrable function that is not Lebesgue integrable on an interval , then there exists a closed and nowhere dense subset such that is not Lebesgue integrable on any subinterval whenever . is called the set of nonsummability points or the set of singularities of the Henstock integral. Furthermore,for some disjoint open subintervals satisfy

In addition, is Lebesgue integrable on any for all . That is, might only have nonsummable points at the limits of the intervals (see [4]). This argument motivates us to consider the local case, particularly when has possible points of singularity at the limit points only.

Remark 1. Without loss of generality, we can assume thatcould only attain a nonsummable point at the right limit. Note that ifhas singularities at both limits, we can split the integral into two integrals using the fact that (see [7])Moreover, if has a singularity at the left limit, then using a change of variables will transfer the singularity to the right limit where we have

Let be a gauge integrable function on with a possible singularity at ; then, the indefinite integralis a continuous function on (see [4]). Therefore, is both gauge and Lebesgue integrable. Integrating by parts in the gauge integral sense yields that

We notice that the integrand on the left-hand side is Lebesgue integrable, while the integrand on the right-hand side is not necessarily Lebesgue integrable (it is gauge integrable). For example, consider the functionon , which we will discuss in Proposition 2. This observation motivates us to study the class of all gauge integrable functions on which at the same time satisfy the condition that is Lebesgue integrable on . So, we start by introducing the following basic proposition.

Proposition 1. Ifis Lebesgue integrable on, then the only possible point of nonsummability foronis.

Proof. It is clear that is Lebesgue integrable on for all . Also, we have in the Lebesgue sense thatThus, is Lebesgue integrable on for all , and hence, the only possible point of nonsummability is .

Let denote as usual the space of all Lebesgue integrable functions on equipped with the norm . In view of the above discussion, we introduce the following subspace of gauge integrable functions.

Definition 6. .

Now, in an obvious way, we define a norm on the space .

Definition 7. .

In the above definition, represents the class of all functions equal to almost everywhere. Indeed, is a norm on since it is a linear combination of norms, and its properties are inherited from the norms and .

Proposition 2. .

Proof. The inclusion follows from the facts that and the uniform boundedness of the term , while the inclusion is clear from the definition of . To see that these inclusions are proper, consider the following example:where . This function satisfies the following properties (see [8]):(i) is not of bounded variation on the interval , and hence, it is not absolutely continuous on .(ii) is differentiable on , andThus, by Theorem 1, is gauge integrable, but it is not Lebesgue integrable. Note that the term is Lebesgue integrable since it is bounded and measurable. Choosing and , the functionis gauge integrable since . Also, it is clear that . In contrast, is not Lebesgue integrable since if it is Lebesgue integrable, and this will imply that is also in , which is a contradiction. Therefore, . To show that is nonempty, choose and , then the functionis gauge integrable, but it is not in since .

Theorem 3. is a complete normed space under the norm.

Proof. Let be a Cauchy sequence on the space . Then, and are Cauchy sequences on the spaces and , respectively. By the completeness of the Lebesgue space, there exists such thatDefine . By Proposition 1, is Lebesgue integrable on for all . Also, we haveHence, converges to in for all , and we getOn the other hand and since is a Cauchy sequence on the space , for all , there exists such thatThat is,Therefore, the sequenceis a uniformly Cauchy sequence on the real line with respect to the variable . Considering in (23) the particular case , we get by the completeness of that there exists satisfyingAlso, for all , the convergence in (21) is uniform. Hence, we can interchange the limits in the following way:Thus, applying Theorem 2 (Hake’s Theorem) and using the result in (25), we getUsing Hake’s theorem again, we conclude that is gauge integrable andWe combine the uniform convergence in (21) and the result in (28) to get thatwhich implies thatAlso, it is clear from the definition of thatTherefore, converges to in , which completes the proof.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.


The author was supported by the Umm Al-Qura University.