# A Complete Normed Space of a Class of Guage Integrable Functions

Authors: Yassin Alzubaidi
Journal: Journal of Mathematics
 (2022)

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## Abstract

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.


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## Body

## 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, a,b will denote a nondegenerate finite closed interval.Definition 1.
A gauge ona,bis a strictly positive functionδ:a,b⟶0,∞.Definition 2.
LetP=xi−1,xi,tii=1nbe a tagged partition ofa,b, and letδbe a gauge ona,b. We say thatPisδ-fine if for alli=1,…,n, and we have(1)xi−1,xi⊂ti−δti,ti+δti.Ifδ is any gauge on a,b, then Cousin’s lemma (see [3]) guarantees the existence of a tagged partition P of a,b, such that P is δ-fine. This result allows us to introduce the following main definition.Definition 3.
A functionf:a,b⟶Ris said to be gauge integrable if there exists a real numberAsuch that for allε>0, there exists a gaugeδεona,bsatisfying the following; ifP=xi−1,xi,tii=1nis a tagged partition ofa,bandPisδε-fine, then(2)∑i=1nftixi−xi−1−A<ε.
In this case, we write(3)∫abfdx=A.It is clear that the gauge integral is a generalization of the Riemann Integral, where the constantδ in Riemann’s definition is being replaced with a positive function (gauge) δt. 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.
LetF:a,b⟶ℝbe a continuous function, which is differentiable nearly everywhere andF′=f, then(i)
fis gauge integrable ona,b(ii)
fis Lebesgue integrable ona,bif and only ifFis absolutely continuousAnother 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.
LetF:a,b⟶ℝbe a function, thenfis gauge integrable ona,bif and only iffis gauge integrable ona,cfor allc∈a,band(4)limc⟶b−∫acf,exists. In this case,(5)∫abf=limc⟶b−∫acf.Definition 4.
Letfbe a gauge integrable function, then(6)fA=supa≤c≤b∫acf.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 f−gA=0⇔f=g a. e, ⋅A is considered a norm once we identify the functions which are equal almost everywhere.Definition 5.
G=Ga,b=f:a,b⟶ℝ:fA<∞G,⋅A 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 Ga,b, 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, iff is a gauge integrable function that is not Lebesgue integrable on an interval I, then there exists a closed and nowhere dense subset D⊂I such that f is not Lebesgue integrable on any subinterval J whenever J°∩D≠ϕ. D is called the set of nonsummability points or the set of singularities of the Henstock integral. Furthermore,(7)∫If=∫Df+∑i=1∞∫Jif,for some disjoint open subintervals Ji satisfy(8)I∖D=∪i=1∞Ji.In addition,f is Lebesgue integrable on any c,d⊂Ji for all i. That is, f might only have nonsummable points at the limits of the intervals Ji (see [4]). This argument motivates us to consider the local case, particularly when f has possible points of singularity at the limit points only.Remark 1.
Without loss of generality, we can assume thatfcould only attain a nonsummable point at the right limit. Note that iffhas singularities at both limits, we can split the integral into two integrals using the fact that (see [7])(9)∫abf=∫acf+∫cbf,forc∈a,b.
Moreover, iff has a singularity at the left limit, then using a change of variables will transfer the singularity to the right limit where we have(10)∫abfx=∫abfa+b−x.Letf be a gauge integrable function on a,b with a possible singularity at b; then, the indefinite integral(11)Fx=∫axf,is a continuous function on a,b (see [4]). Therefore, F is both gauge and Lebesgue integrable. Integrating by parts in the gauge integral sense yields that(12)∫abF=∫ab∫axf=x∫axfab−∫abxf=∫abb−xf.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 function(13)fx=1b−x2cos1b−x3,on a,b, which we will discuss in Proposition 2. This observation motivates us to study the class of all gauge integrable functions on a,b which at the same time satisfy the condition that b−xf is Lebesgue integrable on a,b. So, we start by introducing the following basic proposition.Proposition 1.
Ifb−xfis Lebesgue integrable ona,b, then the only possible point of nonsummability forfona,bisb.Proof.
It is clear thatb−xf is Lebesgue integrable on a,c for all c∈a,b. Also, we have in the Lebesgue sense that(14)∫acf=∫ac1b−xb−xf≤1b−c∫abb−xf<∞.
Thus,f is Lebesgue integrable on a,c for all c∈a,b, and hence, the only possible point of nonsummability is b.LetL=La,b denote as usual the space of all Lebesgue integrable functions on a,b equipped with the norm fL=∫abf. In view of the above discussion, we introduce the following subspace of gauge integrable functions.Definition 6.
G1=G1a,b=f∈Ga,b:b−xf∈La,b.Now, in an obvious way, we define a norm on the spaceG1.Definition 7.
fG1=fA+b−xfL.In the above definition,f represents the class of all functions equal to f almost everywhere. Indeed, ⋅G1 is a norm on G1 since it is a linear combination of norms, and its properties are inherited from the norms ⋅A and ⋅L.Proposition 2.
La,b⊊G1a,b⊊Ga,b.Proof.
The inclusionLa,b⊂G1 follows from the facts that L⊂G and the uniform boundedness of the term b−x, while the inclusion G1⊂G is clear from the definition of G1. To see that these inclusions are proper, consider the following example:(15)Fx=b−xpsin1b−xq,ifx∈a,b,0,ifx=b,where 1≤p≤q. This function F satisfies the following properties (see [8]):(i)
F is not of bounded variation on the interval a,b, and hence, it is not absolutely continuous on a,b.(ii)
F is differentiable on a,b, and(16)F′x=1b−xq+1−pcos1b−xq+pb−xp−1sin1b−xqifx∈a,b.
Thus, by Theorem1, F′ is gauge integrable, but it is not Lebesgue integrable. Note that the term h=pb−xp−1sin1/b−xq is Lebesgue integrable since it is bounded and measurable. Choosing p=3 and q=3, the function(17)f1=1b−xcos1b−x3,is gauge integrable since f1=F′−h. Also, it is clear that b−xf1∈La,b. In contrast, f1 is not Lebesgue integrable since if it is Lebesgue integrable, and this will imply that F′=f1+h is also in La,b, which is a contradiction. Therefore, La,b⊊G1a,b. To show that G∖G1 is nonempty, choose p=1 and q=3, then the function(18)f2=1b−x2cos1b−x3,is gauge integrable, but it is not in G1 since b−xf2=f1∉L.Theorem 3.
G1a,bis a complete normed space under the norm⋅G1.Proof.
Letfn be a Cauchy sequence on the space G1,⋅G1. Then, fn and b−xfn are Cauchy sequences on the spaces G,⋅A and L,⋅L, respectively. By the completeness of the Lebesgue space, there exists g∈L such that(19)limn⟶∞b−xfn=ginL.
Definef=g/b−x. By Proposition 1, f is Lebesgue integrable on a,r for all r∈a,b. Also, we have(20)limn⟶∞∫arfn−f≤limn⟶∞1b−r∫arb−xfn−g=0.
Hence,fn converges to f in La,r for all r∈a,b, and we get(21)∫arf=limn⟶∞∫arfn.
On the other hand and sincefn is a Cauchy sequence on the space G,⋅A, for all ε>0, there exists Nε∈ℕ such that(22)fn−fmA<εfor alln,m>Nε.
That is,(23)supa≤r≤b∫arfn−∫arfm<εfor alln,m>Nε.
Therefore, the sequence(24)∫arfn,is a uniformly Cauchy sequence on the real line ℝ with respect to the variable r. Considering in (23) the particular case r=b, we get by the completeness of ℝ that there exists A∈ℝ satisfying(25)limn⟶∞∫abfn=A.
Also, for allr∈a,b, the convergence in (21) is uniform. Hence, we can interchange the limits in the following way:(26)limr⟶b−∫arf=limr⟶b−limn⟶∞∫arfn=limn⟶∞limr⟶b−∫arfn.
Thus, applying Theorem2 (Hake’s Theorem) and using the result in (25), we get(27)limr⟶b−∫arf=limn⟶∞∫abfn=A.
Using Hake’s theorem again, we conclude thatf is gauge integrable and(28)∫abf=limn⟶∞∫abfn.
We combine the uniform convergence in (21) and the result in (28) to get that(29)supa≤r≤b∫arfn−∫arf⟶0asn⟶∞,which implies that(30)limn⟶∞fn=finG.
Also, it is clear from the definition off that(31)limn⟶∞b−xfn=b−xfinL.
Therefore,fn converges to f in G1, which completes the proof.



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