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

Authors: Tomás Pérez Becerra; Juan Alberto Escamilla Reyna; Daniela Rodríguez Tzompantzi; Jose Jacobo Oliveros Oliveros; Khaing Khaing Aye
Journal: Journal of Function Spaces
 (2018)

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## 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.


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## Body

## 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 Section2 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 paperX, Y, and Z will denote three Banach spaces, ·X, ·Y, and ·Z, which will denote their respective norms, X∗ the dual of X, B:X×Y→Z a bounded bilinear operator fixed, and a,b a closed finite interval of the real line with the usual topology and the Lebesgue measure, which we denote by μ. For a function f:a,b→R we denote the Lebesgue integral of f on a measurable E⊂a,b, when it exists, by L∫E‍f.Definition 1.
B : X × Y → Z is a bounded bilinear operator if B is linear in each variable and there exists M>0 such that Bx,yZ≤MxXyY; in this case, the norm of the operator B is B=inf⁡{M>0:Bx,yZ≤MxXyY}.We say thatP={ti-1,ti,ξi:i=1,…,n} is a tagged partition of a,b if {ti-1,ti:i=1,…,n} is a finite collection of nonoverlapping closed intervals whose union is a,b such that ξi∈[ti-1,ti] for every i. Given a function δ from a,b to 0,∞, called gauge on [a,b], we say that a tagged partition {ti-1,ti,ξi:i=1,…,n} is δ-fine if (1)ti-1,ti⊂ξi-δξi,ξi+δξiforeveryi.Definition 2.
A functionf:a,b→X is Kurzweil integrable in a,b if there exists w∈X such that for every ϵ>0 there exists a gauge δ on [a,b] such that if {ti-1,ti,ξi:i=1,…,n} is a δ-fine tagged partition of a,b, then (2)∑i=1nfξiti-ti-1-wX<ϵ.We write w=K∫ab‍f.Definition 3.
A functionf:a,b→X is Henstock integrable in a,b if there exists F:a,b→X such that for every ϵ>0 there exists a gauge δ on a,b such that if {ti-1,ti,ξi:i=1,…,n} is a δ-fine tagged partition of a,b, then (3)∑i=1nfξiti-ti-1-Fti-Fti-1X<ϵ.We write Fb-Fa=H∫ab‍f.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 F:a,b→X, called primitive, given by Ft=H∫at‍f or F(t)=K∫at‍f, t∈a,b.Definition 4.
LetF:[a,b]→X and let E be a subset of [a,b]. (1)
F is said to be of strongly bounded variation (BV) on E if the number VF,E≔sup∑i‍Fdi-FciX is finite, where the supremum is taken over all finite sequences ci,di of nonoverlapping intervals that have endpoints in E.(2)
F is BV∗ on E if sup∑i‍ωF,ci,di is finite, where the supremum is taken over all finite sequences {[ci,di]} of nonoverlapping intervals that have endpoints in E, and ωF;ci,di=sup{Fy-Fx:x,y∈ci,di} is the oscillation of F on ci,di.(3)
F is said to be strongly absolutely continuous on E or ACE if for every ϵ>0 there exists η>0 such that, for every finite or infinite sequence of nonoverlapping intervals ci,di, with ∑i‍di-ci<η, we have ∑i‍Fdi-Fci≤ϵ where ci,di∈E for all i.(4)
F is AC∗E if for every ϵ>0 there exists η>0 such that, for every finite or infinite sequence or nonoverlapping intervals ci,di satisfying ∑i‍(di-ai)<η, where ci,di∈E for all i, we have ∑i‍ωF;ci,di<ϵ.(5)
F is ACG∗ on E if E is the union of a sequence of closed sets Ei such that, on each Ei, F is AC∗Ei.The next result gives us a characterization of Henstock integrability.Theorem 5 (see [8, Thm. 7.4.5, pp. 217]).
The functionf:a,b→X is Henstock integrable on a,b with the primitive F if and only if F:a,b→X is continuous and ACG∗ on a,b such that F′t=ft almost everywhere (a.e.) in [a,b], where the derivative is in the sense of Frechet.If a functionF:a,b→X is BV or BV∗ on E, then it is bounded on E; that is, K>0 exists such that FtX≤K, for every t∈E. As an ACE function is BV on E and an AC⁡∗E function is BV∗ on E (immediately from the definitions), then every AC⁡E or AC⁡∗E function is also bounded in E. It is easy to see that if F is AC⁡E and E0⊂E, then F is AC⁡E0, similarly if F is AC⁡∗E.The definition of a function of strongly bounded variation can be extended considering the bilinear operatorB:X×Y→Z.Definition 6.
LetG:a,b→Y be a function and D=t0,t1,…,tn a partition of a,b; we define(4)VabG,D=sup∑i=1nBxi,Gti-Gti-1Z,where the supremum is taken over all possible elections of xi∈X, i=1,2,…,n, with xiX≤1. (5)BvarabG=sup⁡VabG,D,where the supremum is taken over all partitions of the interval a,b and BvarabG is the strong B-variation of G on a,b. If we consider (6)V^abG,D=sup⁡∑i=1mBxi,Gtj-Gtj-1Zin the equality (4), then we define Bvar^abG=supV^abG,D as the B-variation of G on [a,b].IfBvarabG<∞ or Bvar^abG<∞ we say that G is of strongly bounded B-variation or G is of bounded B-variation, respectively.It is straightforward that each function of strongly bounded variation is of strongly boundedB-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.
I ∈ Z is the Kurzweil-Stieltjes integral of f:a,b→X with respect to g:a,b→Y if for every ϵ>0 there exists a gauge δ on a,b such that (7)∑i=1nBfξi,gti-gti-1-IZ<ϵ,for every δ-fine tagged partition ti-1,t1,ξi:i=1,…,n of a,b.
In this case we writeI=KS∫ab‍Bf,dg.Now, we introduce the following integral.Definition 8.
A functionf:a,b→X is Henstock-Stieltjes integrable in a,b with respect to g:a,b→Y if there exists H:a,b→Z such that for every ϵ>0 there exists a gauge δ of a,b such that if ti-1,ti,ξi:i=1,…,n is a δ-fine tagged partition of a,b, then (8)∑i=1nBfξi,gti-gti-1-Hti-Hti-1Z<ϵ.We write H(b)-H(a)=HS∫ab‍Bf,dg.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 functionsf, fn:a,b→X, and g:a,b→Y are given. If Bvar^abg<∞, the Kurzweil-Stieltjes integrals KS∫ab‍Bfn,dg exist and the sequence fn converges on a,b uniformly to f, then the integral KS∫ab‍Bf,dg exists and (9)KS∫abBfn,dg=limn→∞KS∫abBf,dg.


### 2.2. Bochner Integral

Let us recall that a functions:a,b→X is called simple if there is a finite sequence {Em}m=1p⊂a,b of Lebesgue measurable sets such that Em∩El=∅ for m≠l and a,b=⋃m=1pEm, where st=xm∈X for t∈Em, m=1,…,p, and in this case the Bochner integral of s is B∫ab‍s=∑m‍xmμEm.A functionf:a,b→X is strongly measurable if there exists a sequence of simple functions that converges pointwise to f a.e. on a,b.A functionf:a,b→X is Bochner integrable if there is a sequence of simple functions fn:a,b→X, n∈N, such that limn→∞fnt=ft a.e. in a,b and (10)limnL∫abfn-f=0;the Bochner integral of f:[a,b]→X is denoted by (B)∫ab‍f and is defined by (11)B∫abf=limnB∫abfn.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 functionf:a,b→X is Bochner integrable on a,b if there exists a function g:a,b→R, which is Lebesgue integrable such that ftX≤gt, t∈a,b.Theorem 11 (see [8, Thm. 7.4.5, pp. 222]).
A functionf:a,b→X is Bochner integrable on a,b if and only if there exists a function F:a,b→X, which is AC on a,b such that F′t=ft a.e. on [a,b].GivenF:a,b→X and g:a,b→Y we define the function BF,g:a,b→Z, given by BF,gt=BFt,gt; we will use this function from now on.Lemma 12.
IfF:a,b→X is a continuous function and g:a,b→Y is Bochner integrable, then the function BF,g is Bochner integrable.Proof.
SinceF is continuous, then it is strongly measurable; moreover there exists K>0 such that FtX≤K for all t∈a,b. Because g is strongly measurable and B is continuous, BF,g is strongly measurable.
By Theorem10 the real function gtY is Lebesgue integrable.
Now(12)BF,gtZ≤BFtXgtY≤KBgtY;hence BF,g 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,x∗F:a,b→R is measurable for each x∗∈X∗; 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)).
LetF:a,b→X be a function. The following conditions are equivalent: (i)
F is strongly measurable.(ii)
F is weakly measurable and there exists a measurable set E⊂a,b with μa,b-E=0 such that FE is separable.Theorem 14.
IfF:a,b→X is continuous a.e., then F is strongly measurable.Proof.
AsF is continuous a.e., then F is weakly continuous; that is, x∗F:a,b→R is continuous a.e. and, hence, measurable.
We defineE=t∈a,b:Ftiscontinuous. E is Lebesgue measurable with μa,b-E=0, as a,b is separable and then E is separable and, furthermore, F is continuous, and then FE is separable; hence F is strongly measurable by the Pettis Theorem.Lemma 15 (see [11, Lemma 6]).
Ifg:a,b→X is of strongly bounded variation on a,b, then g is Bochner integrable on a,b.As a consequence of Lemmas12 and 15, we have the following result.Corollary 16.
Letf:a,b→X be Henstock integrable on a,b and F its primitive, g:a,b→Y of strongly bounded variation, then BF,g is Bochner integrable on a,b.It is easy to prove that the set of functionsAC⁡, AC∗, ACG, and ACG∗ on E⊂a,b form vector spaces with the sum and product by scalars; moreover, these spaces of functions are algebras under the bilinear operator B.Lemma 17.
LetF:a,b→X and G:a,b→X be functions and E a subset of a,b. If F is ACG∗E and G is ACE, then F+G is ACG∗E.Proof.
SinceF is ACG∗E then E=⋃nEn, where F is AC∗En.
For everyϵ>0 there exists η>0 such that, for every {[ci,di]:ci,di∈En}, with ∑i‍di-ci<η, we have ∑i‍ωF;ci,di<ϵ and ∑i‍Gdi-GciX<ϵ, and(13)Fdi+Gdi-Fci-GciX≤Fdi-FciX+Gdi-GciX,and then ∑i‍ωF+G;ci,di<2ϵ.Lemma 18.
LetF:a,b→X and g:a,b→Y be functions and E a subset of a,b. If F is ACG∗E and g is ACE, then BF,g is ACG∗E.Proof.
The proof is analogous to Lemma17 changing the inequality 2 by (14)BFdi,gdi-BFci,gciZ≤BFdi-FciXgdiY+BFciXgdi-gciY.


## 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.
I ∈ Z is the Kurzweil-Stieltjes integral of f:a,b→X with respect to g:a,b→Y if for every ϵ>0 there exists a gauge δ on a,b such that (7)∑i=1nBfξi,gti-gti-1-IZ<ϵ,for every δ-fine tagged partition ti-1,t1,ξi:i=1,…,n of a,b.
In this case we writeI=KS∫ab‍Bf,dg.Now, we introduce the following integral.Definition 8.
A functionf:a,b→X is Henstock-Stieltjes integrable in a,b with respect to g:a,b→Y if there exists H:a,b→Z such that for every ϵ>0 there exists a gauge δ of a,b such that if ti-1,ti,ξi:i=1,…,n is a δ-fine tagged partition of a,b, then (8)∑i=1nBfξi,gti-gti-1-Hti-Hti-1Z<ϵ.We write H(b)-H(a)=HS∫ab‍Bf,dg.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 functionsf, fn:a,b→X, and g:a,b→Y are given. If Bvar^abg<∞, the Kurzweil-Stieltjes integrals KS∫ab‍Bfn,dg exist and the sequence fn converges on a,b uniformly to f, then the integral KS∫ab‍Bf,dg exists and (9)KS∫abBfn,dg=limn→∞KS∫abBf,dg.


## 2.2. Bochner Integral

Let us recall that a functions:a,b→X is called simple if there is a finite sequence {Em}m=1p⊂a,b of Lebesgue measurable sets such that Em∩El=∅ for m≠l and a,b=⋃m=1pEm, where st=xm∈X for t∈Em, m=1,…,p, and in this case the Bochner integral of s is B∫ab‍s=∑m‍xmμEm.A functionf:a,b→X is strongly measurable if there exists a sequence of simple functions that converges pointwise to f a.e. on a,b.A functionf:a,b→X is Bochner integrable if there is a sequence of simple functions fn:a,b→X, n∈N, such that limn→∞fnt=ft a.e. in a,b and (10)limnL∫abfn-f=0;the Bochner integral of f:[a,b]→X is denoted by (B)∫ab‍f and is defined by (11)B∫abf=limnB∫abfn.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 functionf:a,b→X is Bochner integrable on a,b if there exists a function g:a,b→R, which is Lebesgue integrable such that ftX≤gt, t∈a,b.Theorem 11 (see [8, Thm. 7.4.5, pp. 222]).
A functionf:a,b→X is Bochner integrable on a,b if and only if there exists a function F:a,b→X, which is AC on a,b such that F′t=ft a.e. on [a,b].GivenF:a,b→X and g:a,b→Y we define the function BF,g:a,b→Z, given by BF,gt=BFt,gt; we will use this function from now on.Lemma 12.
IfF:a,b→X is a continuous function and g:a,b→Y is Bochner integrable, then the function BF,g is Bochner integrable.Proof.
SinceF is continuous, then it is strongly measurable; moreover there exists K>0 such that FtX≤K for all t∈a,b. Because g is strongly measurable and B is continuous, BF,g is strongly measurable.
By Theorem10 the real function gtY is Lebesgue integrable.
Now(12)BF,gtZ≤BFtXgtY≤KBgtY;hence BF,g 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,x∗F:a,b→R is measurable for each x∗∈X∗; 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)).
LetF:a,b→X be a function. The following conditions are equivalent: (i)
F is strongly measurable.(ii)
F is weakly measurable and there exists a measurable set E⊂a,b with μa,b-E=0 such that FE is separable.Theorem 14.
IfF:a,b→X is continuous a.e., then F is strongly measurable.Proof.
AsF is continuous a.e., then F is weakly continuous; that is, x∗F:a,b→R is continuous a.e. and, hence, measurable.
We defineE=t∈a,b:Ftiscontinuous. E is Lebesgue measurable with μa,b-E=0, as a,b is separable and then E is separable and, furthermore, F is continuous, and then FE is separable; hence F is strongly measurable by the Pettis Theorem.Lemma 15 (see [11, Lemma 6]).
Ifg:a,b→X is of strongly bounded variation on a,b, then g is Bochner integrable on a,b.As a consequence of Lemmas12 and 15, we have the following result.Corollary 16.
Letf:a,b→X be Henstock integrable on a,b and F its primitive, g:a,b→Y of strongly bounded variation, then BF,g is Bochner integrable on a,b.It is easy to prove that the set of functionsAC⁡, AC∗, ACG, and ACG∗ on E⊂a,b form vector spaces with the sum and product by scalars; moreover, these spaces of functions are algebras under the bilinear operator B.Lemma 17.
LetF:a,b→X and G:a,b→X be functions and E a subset of a,b. If F is ACG∗E and G is ACE, then F+G is ACG∗E.Proof.
SinceF is ACG∗E then E=⋃nEn, where F is AC∗En.
For everyϵ>0 there exists η>0 such that, for every {[ci,di]:ci,di∈En}, with ∑i‍di-ci<η, we have ∑i‍ωF;ci,di<ϵ and ∑i‍Gdi-GciX<ϵ, and(13)Fdi+Gdi-Fci-GciX≤Fdi-FciX+Gdi-GciX,and then ∑i‍ωF+G;ci,di<2ϵ.Lemma 18.
LetF:a,b→X and g:a,b→Y be functions and E a subset of a,b. If F is ACG∗E and g is ACE, then BF,g is ACG∗E.Proof.
The proof is analogous to Lemma17 changing the inequality 2 by (14)BFdi,gdi-BFci,gciZ≤BFdi-FciXgdiY+BFciXgdi-gciY.


## 3. Integration by Parts Theorem

### 3.1. Involving Bochner Integral

Theorem 19.
Letf:a,b→X be a Henstock integrable function with primitive F, g:a,b→Y of strongly bounded variation, and G the Bochner primitive of g. Then H∫ab‍Bf,G exists and(15)H∫abBf,G=BFb,Gb-B∫abBF,g.Proof.
By Corollary16, B(F,g) is Bochner integrable. Let ϕ:[a,b]→Z be a function given by (16)ϕt=BFt,Gt-B∫atBF,g.
ϕ is continuous due to the continuity on a,b of F, G, and B∫at‍BF,g.
Theorem5 implies that F is ACG∗ on a,b and Theorem 11 implies that G is AC⁡ on a,b; hence the function t→BFt,Gt is ACG∗ on a,b by Lemma 18. Finally, Lemma 17 implies that ϕ is ACG∗ on a,b.
In order to prove thatϕ is differentiable on a,b, using the fact that BF(t)/s-t,G(s)=BF(t),G(s)/s-t for every s, t∈a,b, we calculate (17)BFs,Gs-BFt,Gts-t-BFt,gt-Bft,GtZ≤BFs-Fts-t-ftXGsY+BFtXGt-Gss-t-gtY+BftXGs-GtY,which tends to 0 when s→t; hence BF,G is differentiable on a,b and BF,G′=BF,g+Bf,G. Then ϕ′t=BF,G′-BF,g=BF,g+Bf,G-BF,g=Bf,G. By Theorem 5, H∫ab‍Bf,G exists and (15) is fulfilled.


### 3.2. Involving Henstock-Stieltjes Integral

Theorem 20.
Letf:a,b→X and g:a,b→Y be functions. If the integral HS∫ab‍Bf,dg exists and Bvarabg<∞, then (18)HS∫abBf,dgZ≤supt∈a,bftXBvarabg.Proof.
Letϵ>0. There exists a gauge δ on a,b such that (19)∑i=1nBfξi,gti-gti-1-HS∫ti-1tiBf,dgZ<ϵ,for every P=ti-1,ti,ξi:i=1,…,n, which is δ-fine. Then, (20)HS∫abBf,dgZ≤HS∫abBf,dg-∑i=1nBfξi,gti-gti-1+∑i=1nBfξi,gti-gti-1Z≤ϵ+∑i=1,fξi≠0nBfξifξiXfξiX,gti-gti-1Z≤ϵ+supt∈a,bftXBvarabg.Now, we shall prove the following Theorem, which is a consequence of Theorem9.Theorem 21 (uniform convergence theorem).
Letf,fn:a,b→X, n=1,2,…, and g:a,b→Y. If (B)varab(g)<∞, the integrals HS∫ab‍Bfn,dg exist for each n, and the sequence fn converges uniformly to f in [a,b], then the integral HS∫ab‍Bf,dg exists and (21)HS∫abBf,dg=limn→∞HS∫abBfn,dg.Proof.
Letϵ>0; because fn converges uniformly to f, there exists n0>0 such that for every n>n0 and t∈a,b(22)fnt-ftX<ϵ.Hence, for every m,n>n0, fnt-fmtX≤2ϵ.
Theorem9 implies the existence of the integral HJ=HS∫J‍Bf,dg for every J⊂a,b and limn→∞HnJ=HJ. Hence, there exists n1∈N such that HnJ-HJZ<ϵ, for every n>n1.
Letm>maxn0,n1 be fixed; as the integral (23)HS∫abBfm,dg≔Hmb-Hmaexists, there exists a gauge δ on a,b such that, for every δ-fine tagged partition ti-1,ti,ξi:i=1,…,n of a,b, (24)∑i=1nBfmξi,gti-gti-1-HS∫ti-1t1Bfm,dgZ<ϵ.We have (25)∑i=1nBfξi,gti-gti-1-Hti-Hti-1Z≤∑i=1nBfξi,gti-gti-1-Bfmξi,gti-gti-1Z+∑i=1nBfmξi,gti-gti-1-HS∫ti-1t1Bfm,dgZ+∑i=1nHS∫ti-1t1Bfm,dg-Hti-Hti-1Z≤∑i,fξi≠fmξiBfξi-fmξifξi-fmξiXfξi-fmξiX,gti-gti-1Z+2ϵ≤supt∈a,bft-fmtXBvarabg+2ϵ≤3ϵ.Hence, HS∫ab‍Bf,dg exists and (26)HS∫abBf,dg=limn→∞HS∫abBfn,dg.Theorem 22.
Iff:a,b→X is a step function, then for every g:a,b→Y, the integral HS∫ab‍Bf,dg exists.Proof.
Analogous to the proof of [12, Lemma 3.2], is enough to prove for functions of the forms χa,τx, χτ,bx, χax, and χbx, where τ∈a,b and x∈X. Let τ∈a,b, x∈X, and f=χa,τx. Given ϵ>0 we define δ(t)=ϵ if t=τ and δt=1/2τ-t if t≠τ; then for any δ-fine tagged partition {([ti-1,ti],ξi):i=1,…,n} of a,b, τ is the tag of one subinterval, if ξi∈τ,b, fξi=0, and ∫τb‍Bf,dg=0; otherwise (27)∑i=1nBfξi,gti-gti-1-Bx,gti-Bx,gti-1Z=∑i=1nBfξi-x,gti-gti-1Z=0<ϵ.Hence HS∫ab‍Bf,dg=Bx,gτ-Bx,ga. The proofs of the cases f=χτ,bx, f=χax, and f=χbx 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]).
F : a , b → X is regulated if and only if it is the uniform limit of step functions.Theorem 24.
IfF:a,b→X is regulated and g:a,b→Y with Bvarabg<∞, then the integral HS∫ab‍BF,dg exists.Proof.
In as much asF is regulated, there exists a sequence Fn:a,b→X, n=1,2,…, of step functions which converges uniformly to F, by Theorem 22; the integrals HS∫ab‍BFn,dg exist for each n=1,2,…. The Uniform Convergence Theorem implies the existence of the integral HS∫ab‍BF,dg.Theorem 25 (integration by parts theorem).
Iff:a,b→X is Henstock integrable, F its primitive, and g:a,b→Y with Bvarabg<∞; then H∫ab‍Bf,g exists and(28)H∫abBf,g=BFb,gb-HS∫abBF,dg.Proof.
Letϵ>0. Since f is Henstock integrable, with F its primitive, there exists a gauge δ1 such that if P1=ti-1,ti,ξi;i=1,2,…,n is a δ1-fine tagged partition of a,b, (29)∑i=1nfξiti-ti-1-Fti-Fti-1X<ϵ.On the other hand, since F is continuous, it is regulated, and by Theorem 24, HS∫ab‍BF,dg exists; then there exists a gauge δ2 such that if P2={sj-1,sj,ηj;j=1,2,…,m} is a δ2-fine tagged partition of a,b, (30)∑j=1mBFηj,gtj-gtj-1-HS∫sj-1sjBF,dgZ<ϵ.We define a gauge δ∗ by δ∗(t)=minδ1t,δ2t. Let P∗=tk-1,tk,ξk,k=i,…,l be a δ∗-fine tagged partition of a,b 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 (31)∑k=1lBftk,gtktk-tk-1-BFtk,gtk-BFtk-1,gtk-1-HS∫tk-1kiBF,dgZ≤∑k=1lBftktk-tk-1-Ftk-Ftk-1,gtkZ+∑k=1lBFtk,gtk-gtk-1-HS∫tk-1tkBF,dgZ≤ϵBvarabg+ϵ.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 Theorem19 is the derivative of the primitive of the function t→Bft,Gt,that is, the Fundamental Theorem of Calculus.(2)
In Theorem25 the Fundamental Theorem of Calculus does not apply because g is not necessarily differentiable, and if it is, the primitive of g′, in general, is not g.(3)
The condition of differentiability on a functiong:a,b→Y of strongly bounded variation is equivalent to Y and has the Radon-Nikodým property (see Distel and Uhl [10, Chapter VII §6]). Is it fulfilled with functions of strongly bounded B-variation?Therefore, the condition ofg is one that ensures the Fundamental Theorem of Calculus, so we have the following theorems.Theorem 26.
LetF:a,b→X be continuous function and g:a,b→Y with Bvarabg<∞ such that g′ exists on a,b. Then (32)H∫abBF,g′=HS∫abBF,dg.Proof.
H S ∫ a b ‍ B F , d g exists by Theorem 24; then given ϵ>0, we have a gauge δ and ti-1,ti,ξi:i=1,…,n a δ-fine tagged partition of a,b. Since g′ exists, for every t∈a,b there exists ηt>0 such that if t-s<ηt then (33)gt-gst-s-g′t<ϵ.We define a gauge δ1:a,b→0,∞ by δ1t=minηt,δt. For all tj-1,tj,ξj:j=1,…,m which is δ1-fine tagged partition of a,b and supposing that each tag is the left endpoint of its respective subinterval we have (34)∑j=1mBFti,g′titi-ti-1-HS∫ti-1tiBF,dgZ≤∑j=1mBFti,g′ti-gti-gti-1ti-ti-1ti-ti-1Z+∑j=1mBFti,gti-gti-1-HS∫ti-1tiBF,dgZ<ϵMBb-a+1,where M>0 is the bound of F due to it is continuous.Obviously, we can change the condition overg in Theorem 26 if we ask for strongly bounded variation and impose the Radon-Nikodým property on Y; either with these conditions or with those of Theorem 26 we can write equality (28) as(35)H∫abBf,g=BFb,gb-H∫abBF,g′.A functionG:a,b→R satisfies the Lipschitz condition if there exists L>0 such that Gt-Gs<Lt-s, s, t∈a,b, and G is of bounded slope variation (BSV) if (36)∑i=0n-2Gti+2-Gti+1ti+2-ti+1-Gti+1-Gtiti+1-tiis bounded for all divisions a=t0<t1<⋯<tn=b.Lee in [14, p. 75] proves that a function G:a,b→R is the primitive (in the sense of Henstock) of a function of strongly bounded variation on a,b if and only if G satisfies the Lipschitz condition and is of bounded slope variation on a,b; 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.
Letf:a,b→X be a function and F its primitive. If g:a,b→Y satisfies the Lipschitz condition and is of bounded slope variation, then H∫ab‍Bf,g exists and (37)H∫abBf,g=BF,gab-H∫abBF,g′.


## 3.1. Involving Bochner Integral

Theorem 19.
Letf:a,b→X be a Henstock integrable function with primitive F, g:a,b→Y of strongly bounded variation, and G the Bochner primitive of g. Then H∫ab‍Bf,G exists and(15)H∫abBf,G=BFb,Gb-B∫abBF,g.Proof.
By Corollary16, B(F,g) is Bochner integrable. Let ϕ:[a,b]→Z be a function given by (16)ϕt=BFt,Gt-B∫atBF,g.
ϕ is continuous due to the continuity on a,b of F, G, and B∫at‍BF,g.
Theorem5 implies that F is ACG∗ on a,b and Theorem 11 implies that G is AC⁡ on a,b; hence the function t→BFt,Gt is ACG∗ on a,b by Lemma 18. Finally, Lemma 17 implies that ϕ is ACG∗ on a,b.
In order to prove thatϕ is differentiable on a,b, using the fact that BF(t)/s-t,G(s)=BF(t),G(s)/s-t for every s, t∈a,b, we calculate (17)BFs,Gs-BFt,Gts-t-BFt,gt-Bft,GtZ≤BFs-Fts-t-ftXGsY+BFtXGt-Gss-t-gtY+BftXGs-GtY,which tends to 0 when s→t; hence BF,G is differentiable on a,b and BF,G′=BF,g+Bf,G. Then ϕ′t=BF,G′-BF,g=BF,g+Bf,G-BF,g=Bf,G. By Theorem 5, H∫ab‍Bf,G exists and (15) is fulfilled.


## 3.2. Involving Henstock-Stieltjes Integral

Theorem 20.
Letf:a,b→X and g:a,b→Y be functions. If the integral HS∫ab‍Bf,dg exists and Bvarabg<∞, then (18)HS∫abBf,dgZ≤supt∈a,bftXBvarabg.Proof.
Letϵ>0. There exists a gauge δ on a,b such that (19)∑i=1nBfξi,gti-gti-1-HS∫ti-1tiBf,dgZ<ϵ,for every P=ti-1,ti,ξi:i=1,…,n, which is δ-fine. Then, (20)HS∫abBf,dgZ≤HS∫abBf,dg-∑i=1nBfξi,gti-gti-1+∑i=1nBfξi,gti-gti-1Z≤ϵ+∑i=1,fξi≠0nBfξifξiXfξiX,gti-gti-1Z≤ϵ+supt∈a,bftXBvarabg.Now, we shall prove the following Theorem, which is a consequence of Theorem9.Theorem 21 (uniform convergence theorem).
Letf,fn:a,b→X, n=1,2,…, and g:a,b→Y. If (B)varab(g)<∞, the integrals HS∫ab‍Bfn,dg exist for each n, and the sequence fn converges uniformly to f in [a,b], then the integral HS∫ab‍Bf,dg exists and (21)HS∫abBf,dg=limn→∞HS∫abBfn,dg.Proof.
Letϵ>0; because fn converges uniformly to f, there exists n0>0 such that for every n>n0 and t∈a,b(22)fnt-ftX<ϵ.Hence, for every m,n>n0, fnt-fmtX≤2ϵ.
Theorem9 implies the existence of the integral HJ=HS∫J‍Bf,dg for every J⊂a,b and limn→∞HnJ=HJ. Hence, there exists n1∈N such that HnJ-HJZ<ϵ, for every n>n1.
Letm>maxn0,n1 be fixed; as the integral (23)HS∫abBfm,dg≔Hmb-Hmaexists, there exists a gauge δ on a,b such that, for every δ-fine tagged partition ti-1,ti,ξi:i=1,…,n of a,b, (24)∑i=1nBfmξi,gti-gti-1-HS∫ti-1t1Bfm,dgZ<ϵ.We have (25)∑i=1nBfξi,gti-gti-1-Hti-Hti-1Z≤∑i=1nBfξi,gti-gti-1-Bfmξi,gti-gti-1Z+∑i=1nBfmξi,gti-gti-1-HS∫ti-1t1Bfm,dgZ+∑i=1nHS∫ti-1t1Bfm,dg-Hti-Hti-1Z≤∑i,fξi≠fmξiBfξi-fmξifξi-fmξiXfξi-fmξiX,gti-gti-1Z+2ϵ≤supt∈a,bft-fmtXBvarabg+2ϵ≤3ϵ.Hence, HS∫ab‍Bf,dg exists and (26)HS∫abBf,dg=limn→∞HS∫abBfn,dg.Theorem 22.
Iff:a,b→X is a step function, then for every g:a,b→Y, the integral HS∫ab‍Bf,dg exists.Proof.
Analogous to the proof of [12, Lemma 3.2], is enough to prove for functions of the forms χa,τx, χτ,bx, χax, and χbx, where τ∈a,b and x∈X. Let τ∈a,b, x∈X, and f=χa,τx. Given ϵ>0 we define δ(t)=ϵ if t=τ and δt=1/2τ-t if t≠τ; then for any δ-fine tagged partition {([ti-1,ti],ξi):i=1,…,n} of a,b, τ is the tag of one subinterval, if ξi∈τ,b, fξi=0, and ∫τb‍Bf,dg=0; otherwise (27)∑i=1nBfξi,gti-gti-1-Bx,gti-Bx,gti-1Z=∑i=1nBfξi-x,gti-gti-1Z=0<ϵ.Hence HS∫ab‍Bf,dg=Bx,gτ-Bx,ga. The proofs of the cases f=χτ,bx, f=χax, and f=χbx 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]).
F : a , b → X is regulated if and only if it is the uniform limit of step functions.Theorem 24.
IfF:a,b→X is regulated and g:a,b→Y with Bvarabg<∞, then the integral HS∫ab‍BF,dg exists.Proof.
In as much asF is regulated, there exists a sequence Fn:a,b→X, n=1,2,…, of step functions which converges uniformly to F, by Theorem 22; the integrals HS∫ab‍BFn,dg exist for each n=1,2,…. The Uniform Convergence Theorem implies the existence of the integral HS∫ab‍BF,dg.Theorem 25 (integration by parts theorem).
Iff:a,b→X is Henstock integrable, F its primitive, and g:a,b→Y with Bvarabg<∞; then H∫ab‍Bf,g exists and(28)H∫abBf,g=BFb,gb-HS∫abBF,dg.Proof.
Letϵ>0. Since f is Henstock integrable, with F its primitive, there exists a gauge δ1 such that if P1=ti-1,ti,ξi;i=1,2,…,n is a δ1-fine tagged partition of a,b, (29)∑i=1nfξiti-ti-1-Fti-Fti-1X<ϵ.On the other hand, since F is continuous, it is regulated, and by Theorem 24, HS∫ab‍BF,dg exists; then there exists a gauge δ2 such that if P2={sj-1,sj,ηj;j=1,2,…,m} is a δ2-fine tagged partition of a,b, (30)∑j=1mBFηj,gtj-gtj-1-HS∫sj-1sjBF,dgZ<ϵ.We define a gauge δ∗ by δ∗(t)=minδ1t,δ2t. Let P∗=tk-1,tk,ξk,k=i,…,l be a δ∗-fine tagged partition of a,b 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 (31)∑k=1lBftk,gtktk-tk-1-BFtk,gtk-BFtk-1,gtk-1-HS∫tk-1kiBF,dgZ≤∑k=1lBftktk-tk-1-Ftk-Ftk-1,gtkZ+∑k=1lBFtk,gtk-gtk-1-HS∫tk-1tkBF,dgZ≤ϵBvarabg+ϵ.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 Theorem19 is the derivative of the primitive of the function t→Bft,Gt,that is, the Fundamental Theorem of Calculus.(2)
In Theorem25 the Fundamental Theorem of Calculus does not apply because g is not necessarily differentiable, and if it is, the primitive of g′, in general, is not g.(3)
The condition of differentiability on a functiong:a,b→Y of strongly bounded variation is equivalent to Y and has the Radon-Nikodým property (see Distel and Uhl [10, Chapter VII §6]). Is it fulfilled with functions of strongly bounded B-variation?Therefore, the condition ofg is one that ensures the Fundamental Theorem of Calculus, so we have the following theorems.Theorem 26.
LetF:a,b→X be continuous function and g:a,b→Y with Bvarabg<∞ such that g′ exists on a,b. Then (32)H∫abBF,g′=HS∫abBF,dg.Proof.
H S ∫ a b ‍ B F , d g exists by Theorem 24; then given ϵ>0, we have a gauge δ and ti-1,ti,ξi:i=1,…,n a δ-fine tagged partition of a,b. Since g′ exists, for every t∈a,b there exists ηt>0 such that if t-s<ηt then (33)gt-gst-s-g′t<ϵ.We define a gauge δ1:a,b→0,∞ by δ1t=minηt,δt. For all tj-1,tj,ξj:j=1,…,m which is δ1-fine tagged partition of a,b and supposing that each tag is the left endpoint of its respective subinterval we have (34)∑j=1mBFti,g′titi-ti-1-HS∫ti-1tiBF,dgZ≤∑j=1mBFti,g′ti-gti-gti-1ti-ti-1ti-ti-1Z+∑j=1mBFti,gti-gti-1-HS∫ti-1tiBF,dgZ<ϵMBb-a+1,where M>0 is the bound of F due to it is continuous.Obviously, we can change the condition overg in Theorem 26 if we ask for strongly bounded variation and impose the Radon-Nikodým property on Y; either with these conditions or with those of Theorem 26 we can write equality (28) as(35)H∫abBf,g=BFb,gb-H∫abBF,g′.A functionG:a,b→R satisfies the Lipschitz condition if there exists L>0 such that Gt-Gs<Lt-s, s, t∈a,b, and G is of bounded slope variation (BSV) if (36)∑i=0n-2Gti+2-Gti+1ti+2-ti+1-Gti+1-Gtiti+1-tiis bounded for all divisions a=t0<t1<⋯<tn=b.Lee in [14, p. 75] proves that a function G:a,b→R is the primitive (in the sense of Henstock) of a function of strongly bounded variation on a,b if and only if G satisfies the Lipschitz condition and is of bounded slope variation on a,b; 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.
Letf:a,b→X be a function and F its primitive. If g:a,b→Y satisfies the Lipschitz condition and is of bounded slope variation, then H∫ab‍Bf,g exists and (37)H∫abBf,g=BF,gab-H∫abBF,g′.


## 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.
LetHa,b,X be the space of all Henstock integrable functions from a,b to X. We define a norm on Ha,b,X, called Alexiewicz norm (see e.g., [5] or [16]), by (38)fA=sup⁡H∫atfX:a≤t≤b.Theorem 29.
Letg:a,b→Y be a function. If Bvarabg<∞, then (39)Tf=H∫abBf,gdefines a continuous linear operator on the space of Henstock integrable functions into Z with (40)T≤BgbY+Bvarabg.Proof.
LetF:a,b→X the primitive of f and ϵ>0; by Theorem 24 there exists a gauge δ such that if ti-1,ti,ξi;i=1,…,n is δ-fine, (41)∑i=1nBFξi,gti-gti-1-HS∫abBF,dgZ<ϵ.
By the integration by parts theorem we have(42)TfZ=H∫abBf,gZ=BFb,gb-HS∫abBF,dgZ≤BFb,gb-∑i=1nBFξi,gti-gti-1Z+∑i=1nBFξi,gti-gti-1-HS∫abBF,dgZ≤BFb,gbZ+∑i=1nBFξi,gti-gti-1Z+ϵ≤BFb,gbZ+fA∑i=1,FξiX≠0nBFξiFξiX,gti-gti-1Z+ϵ≤fABgbY+fABvarabg+ϵ.
It follows that(43)TffA≤BgbY+Bvarabg.So T is bounded and continuous.L X , Y denotes the space of continuous linear operators from X to Y and B:X×LX,Y→Y is the bilinear bounded operator given by Bx,A=Ax.Theorem 30.
LetT:Ha,b,X→Y be a linear continuous operator defined on the space of the Henstock integrable functions. There exists a function g:a,b→LX,Y with Bvarabg<∞ such that (44)Tf=H∫abBf,g,for every f∈Ha,b,X.Proof.
For eacht∈a,b and x∈X we have χa,tx∈Ha,b,X. We define a function βt:X→Y by (45)βtx=Tχa,tx.In this form, βt is linear and it is also continuous due to (46)βtxY=Tχa,txY≤MxX,where M=Tb-a.
We define a functiong:a,b→LX,Y by (47)gt=βt.
Now,Bvarabg<∞, indeed, for every arbitrary partition ti-1,ti:i=1,…,n and for every xi, i=1,…,n, with xiX≤1, (48)∑i=1nBxi,gti-gti-1Y=∑i=1nβtixi-βti-1xiY=∑i=1nTχa,tixi-Tχa,ti-1xiY=∑i=1nTχti-1,tixiY≤T∑i=1nxiXti-ti-1≤Tb-a.
Suppose thats:a,b→X is a step function; by Theorem 22 we have that the integral HS∫ab‍Bs,dg exists.
Givenf∈Ha,b,X, we take its primitive F:a,b→X. Since F is continuous, there exists Fn sequence of linear piecewise functions such that Fn→F uniformly on a,b. The integral HS∫ab‍BFn,dg exists for every n by Theorem 24.
By the Uniform Convergence Theorem, the integralHS∫ab‍BF,dg exists; hence (49)TF=limn→∞TFn=limn→∞HS∫abBFn,dg=HS∫abBF,dg.
Letfn be the sequence of derivatives of Fn; fn is simple for each n, since Fn→F uniformly (50)fn-fA=Fn-F∞⟶0;by the continuity of T and the Integration by parts Theorem 25 we have that (51)Tf=limn→∞Tfn=limn→∞H∫abBfn,g=H∫abBf,g.Finally, we have the following representation theorem.Corollary 31.
T : H a , b , X → Y is a linear continuous operator if and only if there exists a function g:[a,b]→LX,Y with Bvarabg<∞ such that (52)Tf=H∫abBf,g,for every f∈Ha,b,X.ConsiderX a Banach space and Y=X∗ in the definition of strongly bounded B-variation function; then this is equivalent to the definition of strongly bounded variation function, so we have the next result.Corollary 32.
T is a linear continuous functional on Ha,b,X if and only if there exists a function g:a,b→X∗ and Vg,a,b<∞ such that (53)Tf=H∫abgt∘ft,for every f∈Ha,b,X.Hence, the dual space ofHa,b,X 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 Corollary31 (with X=LX,Y=Y=R) we have obtained the following representation theorem for the space of Henstock-Kurzweil integrable functions.Corollary 33.
T : H K a , b , R → R is a linear continuous functional if and only if exists a function g:a,b→R, Vga,b,<∞, such that (54)TF=HK∫abfg,for every f 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 Corollary27 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.
Ifg:a,b→Y is of strongly bounded B-variation and differentiable and g′ is bounded, then T(f)=H∫ab‍Bf,g is a continuous linear operator on the space Ha,b,X into Z.Ifg:a,b→Y 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.
LetT:Ha,b,X→Y be a linear continuous operator defined on the space of the Henstock integrable functions. There exists a function g:a,b→LX,Y of bounded slope variation and Lipschitz such that Tf=H∫ab‍Bf,g, for every f∈Ha,b,X.Proof.
Letx∈X and F be the primitive of f∈Ha,b,X. We define the function βtx=Tχa,tx; this function is in LX,Y, and we define g:a,b→LX,Y by gt=βt; g is of bounded slope variation and satisfies the Lipschitz condition (see the proof of [15, Thm. 10]); hence it is differentiable and g′ is of strongly bounded variation. Following the proof of Theorem 30 using g which is also of strongly bounded variation, by Theorem 26 the integral HS∫ab‍BF,dg is equal to H∫ab‍BF,g′ and by integration by parts Corollary 27, we have Tf=H∫ab‍Bf,g.The last representation theorem identifiesHa,b,X 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 B-variation.



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