EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 13, No. 1, 2020, 130-143
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Simple Properties and Existence Theorem for the
Henstock–Kurzweil–Stieltjes Integral of Functions
Taking Values on C[a, b] Space-valued Functions

Andrew Felix Cunanan1,∗, Julius Benitez2

1 Department of Natural Sciences and Mathematics, College of Arts and Sciences,
Surigao Del Sur State University, Tandag City, Surigao del Sur, Philippines
2 Department of Mathematics and Statistics, College of Sciences and Mathematics,
Mindanao State University-Iligan Institute of Technology, Tibanga, Iligan City, Philippines

Abstract. Henstock–Kurzweil integral, a nonabsolute integral, is a natural extension of the Rie-
mann integral that was studied independently by Ralph Henstock and Jaroslav Kurzweil. This
paper will introduce the Henstock–Kurzweil–Stieltjes integral of C[a, b]-valued functions defined on
a closed interval [f, g] ⊆ C[a, b], where C[a, b] is the space of all continuous real-valued functions
defined on [a, b] ⊆ R. Some simple properties of this integral will be formulated including the
Cauchy criterion and an existence theorem will be provided.

2020 Mathematics Subject Classifications: 58C06,51M20, 26A42, 26B05, 26B30

Key Words and Phrases: C[a, b] space-valued function, δ-fine tagged division, Henstock–Kurzweil–
Stieltjes integral, Continuity, Bounded variation.

1. Introduction

The Henstock–Kurzweil–Stieltjes integral is a generalized Riemann–Stieltjes integral
which has properties similar to it. In the paper [9], Ubaidillah introduce the Henstock–
Kurzweil integral of functions taking values in C[a, b] through Riemann sums

S(F,D) =
∑
D

F (ti)[hi−1, hi]

where D = {([hi−1, hi], ti)}ni=1 is a tagged division of [f, g] Notion of integrals for Ba-
nach space-valued functions like Henstock integral for Banach space-valued functions,
Henstock–Stieltjes integral of real-valued functions with respect to an increasing func-
tion and Henstock–Stieltjes integral for Banach spaces were already defined by Cao [3],

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v13i1.3626

Email addresses: deofscunananiv@yahoo.com (A. Cunanan),
julius.benitez@g.msuiit,edu.ph (J. Benitez)

http://www.ejpam.com 130 c© 2020 EJPAM All rights reserved.



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 131

Lim [7] and Tikare [8], respectively. In this paper we change the way to define the do-
main of the function and the integrator. We shall choose first a closed interval [f, g] as
our domain and a continuous real-valued function H instead of the identity map as our
integrator.

2. Preliminaries

Throughout, we consider the space C[a, b] of all continuous real-valued functions de-
fined on [a, b]. For more details of the space C[a, b], see [2], [5] or [9].

Let [f, g] be a closed interval of C[a, b]. A division of [f, g] is any finite set {h0, h1, . . .
, hn} ⊂ [f, g] such that

h0 = f, hn = g and hi−1 < hi

for all i = 1, 2, . . . , n. A tagged division of [f, g] is a finite collection {([hi−1, hi], ti) :
i = 1, 2, . . . , n} of interval−point pairs such that {h0, h1, . . . , hn} is a division of [f, g]
and ti ∈ [hi−1, hi] for every i = 1, 2, . . . , n. Each point ti is referred to as the tag of the
corresponding subinterval [hi−1, hi]. Let θ be the null element in C[a, b], that is, θ(x) = 0,
for all x ∈ [a, b]. A function δ : [f, g] → C[a, b] is said to be a gauge on [f, g] if θ < δ(h)
for every h ∈ [f, g].

Definition 1. [9] Let δ be a gauge on [f, g]. A tagged division D = {([hi−1, hi], ti) : i =
1, 2, . . . , n} is said to be δ-fine if

ti ∈ [hi−1, hi] ⊂ (ti − δ(ti), ti + δ(ti))

for every i = 1, 2, . . . , n.

Theorem 1. [9] (Cousin’s Lemma) If δ is a gauge on [f, g] ⊂ C[a, b], then there is a δ-fine
tagged division of [f, g].

3. Henstock-Kurzweil-Stieltjes Integral on C[a, b]

Let D = {([hi−1, hi], ti) : i = 1, 2, . . . , n} be a tagged division of [f, g] and F,H :
[f, g]→ C[a, b] be functions. We write

S(F,H;D) =

n∑
i=1

F (ti)[H(hi)−H(hi−1)],

called as Henstock−Kurzweil−Stieltjes sum of F with respect to H on [f, g]. For
brevity, we write D = {([u, v], t)} for a tagged division of [f, g] and

S(F,H;D) =
∑
D

F (t)[H(v)−H(u)].



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 132

Definition 2. Let F,H : [f, g] → C[a, b] be functions. We say that the function F is
Henstock−Kurzweil−Stieltjes integrable with respect to H on [f, g] to S ∈ C[a, b],
briefly HKS-integrable, if for any ε > 0, there exists a gauge δ on [f, g] such that for any
δ-fine tagged division D of [f, g], we have

|S(F,H;D)− S| < ε · e,

where e is the multiplicative identity in C[a, b]. The element S ∈ C[a, b] is called
Henstock−Kurweil−Stieltjes integral, briefly HKS-integral, of F with respect to H
on [f, g] and is written by

S = (HKS)

∫ g

f
F dH.

The collection of all functions which are HKS-integrable with respect to H on [f, g] is
denoted by HKS([f, g], H).

Theorem 2. (Uniqueness) If F is HKS-integrable with respect to H on [f, g], then the
HKS-integral of F with respect to H on [f, g] is unique.

Proof. Suppose that F is HKS-integrable with respect to H on [f, g] to S1 ∈ C[a, b]
and S2 ∈ C[a, b]. Let ε > 0. Then there exists a gauge δ1 on [f, g] such that for all δ1-fine
tagged division D = {([hi−1, hi], ti) : i = 1, 2, . . . , n} of [f, g], we have

|S(F,H;D)− S1| <
ε

2
· e. (1)

Similarly, there exists a gauge δ2 on [f, g] such that for all δ2-fine tagged division Q =
{([ki−1, ki], si) : i = 1, 2, . . . ,m} of [f, g], we have

|S(F,H;Q)− S2| <
ε

2
· e. (2)

Define a function δ : [f, g]→ C[a, b] by δ = δ1 ∧ δ2. Hence, by (1) and (2)

|S1 − S2| <
ε

2
· e+

ε

2
· e = ε · e.

This shows that S1 = S2. Therefore, the HKS-integral of F with respect to H on [f, g] is
unique.

4. Simple Properties

Theorem 3. If F,G ∈ HKS([f, g], H) and α ∈ R, then

(i) Homogenity: α · F ∈ HKS([f, g], H) and

(HKS)

∫ g

f
(α · F )dH = α · (HKS)

∫ g

f
FdH.



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 133

(ii) Linearity: F +G ∈ HKS([f, g], H) and

(HKS)

∫ g

f
(F +G)dH = (HKS)

∫ g

f
FdH + (HKS)

∫ g

f
GdH.

Proof.

(i) Let ε > 0. Then there exists a gauge δ on [f, g] such that for any δ-fine tagged
division D = {([hi−1, hi], ti) : i = 1, 2, . . . , n} of [f, g], we have∣∣∣∣∣

n∑
i=1

F (ti)[H(hi)−H(hi−1)]− (HKS)

∫ g

f
FdH

∣∣∣∣∣ < ε

|α|+ 1
· e.

Thus, for any δ-fine tagged division D = {([hi−1, hi], ti) : i = 1, 2, . . . , n} of [f, g]∣∣∣∣ n∑
i=1

(α · F )(ti)[H(hi)−H(hi−1)]− α · (HKS)

∫ g

f
FdH

∣∣∣∣
=

∣∣∣∣α{ n∑
i=1

F (ti)[H(hi)−H(hi−1)]− (HKS)

∫ g

f
FdH

}∣∣∣∣
= |α| ·

∣∣∣∣ n∑
i=1

F (ti)[H(hi)−H(hi−1)]− (HKS)

∫ g

f
FdH

∣∣∣∣
< |α| · ε

|α|+ 1
· e

< ε · e.

This shows that α · F ∈ HKS([f, g], H) and

(HKS)

∫ g

f
(α · F )dH = α · (HKS)

∫ g

f
FdH.

(ii) Let ε > 0. Then there exists gauge δF on [f, g] such that for any δF -fine tagged
division D = {([hi−1, hi], ti) : i = 1, 2, . . . , n} of [f, g], we have∣∣∣∣∣

n∑
i=1

F (ti)[H(hi)−H(hi−1)]− (HKS)

∫ g

f
FdH

∣∣∣∣∣ < ε

2
· e. (3)

Similarly, there exists gauge δG on [f, g] such that for any δG-fine tagged division
Q = {([ki−1, ki], si) : i = 1, 2, . . . ,m} of [f, g], we have∣∣∣∣∣

m∑
i=1

G(si)[H(ki)−H(ki−1)]− (HKS)

∫ g

f
GdH

∣∣∣∣∣ < ε

2
· e. (4)



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 134

Define δ = δF ∧ δG. Then δ is a gauge on [f, g]. Let D = {([hi−1, hi], ti) : i =
1, 2, . . . , n} be a δ-fine tagged division of [f, g]. Then D is both δF and δG-fine. By
(3) and (4),∣∣∣∣ n∑

i=1

(F +G)(ti)[H(hi)−H(hi−1)]−
{

(HKS)

∫ g

f
FdH + (HKS)

∫ g

f
GdH

}∣∣∣∣
≤
∣∣∣∣ n∑
i=1

F (ti)[H(hi)−H(hi−1)]− (HKS)

∫ g

f
FdH

∣∣∣∣
+

∣∣∣∣ n∑
i=1

G(ti)[H(hi)−H(hi−1)]− (HKS)

∫ g

f
GdH

∣∣∣∣
<
ε

2
· e+

ε

2
· e = ε · e.

Therefore, F +G ∈ HKS([f, g], H) and

(HKS)

∫ g

f
(F +G)dH = (HKS)

∫ g

f
FdH + (HKS)

∫ g

f
GdH.

Theorem 4. (Linearity of Integrator) If F ∈ HKS([f, g], H1) ∩ HKS([f, g], H2), then
F ∈ HKS([f, g], H1 +H2) and

(HKS)

∫ g

f
Fd(H1 +H2) = (HKS)

∫ g

f
FdH1 + (HKS)

∫ g

f
FdH2.

Proof. Let ε > 0. Then there exists gauge δH1 on [f, g] such that for any δH1-fine
tagged division D = {([hi−1, hi], ti) : i = 1, 2, . . . , n} of [f, g], we have∣∣∣∣∣

n∑
i=1

F (ti)[H1(hi)−H1(hi−1)]− (HKS)

∫ g

f
FdH1

∣∣∣∣∣ < ε

2
· e. (5)

Similarly, there exists gauge δH2 on [f, g] such that for any δH2-fine tagged division Q =
{([ki−1, ki], si) : i = 1, 2, . . . ,m} of [f, g], we have∣∣∣∣∣

m∑
i=1

F (si)[H2(ki)−H2(ki−1)]− (HKS)

∫ g

f
FdH2

∣∣∣∣∣ < ε

2
· e. (6)

Define δ = δH1 ∧ δH2 . Then δ is a gauge on [f, g]. Let D = {([hi−1, hi], ti) : i = 1, 2, . . . , n}
be a δ-fine tagged division of [f, g]. Then D is both δH1 and δH2-fine. By (5) and (6),∣∣∣∣ n∑
i=1

F (ti)[(H1 +H2)(hi)− (H1 +H2)(hi−1)]−
{

(HKS)

∫ g

f
FdH1 + (HKS)

∫ g

f
FdH2

}∣∣∣∣



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 135

≤
∣∣∣∣ n∑
i=1

F (ti)[H1(hi)−H1(hi−1)]− (HKS)

∫ g

f
FdH1

∣∣∣∣
+

∣∣∣∣ n∑
i=1

F (ti)[H2(hi)−H2(hi−1)]− (HKS)

∫ g

f
FdH2

∣∣∣∣
<
ε

2
· e+

ε

2
· e = ε · e.

Therefore, F ∈ HKS([f, g], H1 +H2) and

(HKS)

∫ g

f
Fd(H1 +H2) = (HKS)

∫ g

f
FdH1 + (HKS)

∫ g

f
FdH2.

Theorem 5. (Additivity) Let f ≤ r ≤ g. If F ∈ HKS([f, r], H) and F ∈ HKS([r, g], H),
then F ∈ HKS([f, g], H) and

(HKS)

∫ g

f
FdH = (HKS)

∫ r

f
FdH + (HKS)

∫ g

r
FdH.

Proof. Let ε > 0. Then there exists gauge δ1 on [f, r] such that for any δ1-fine tagged
division D = {([hi−1, hi], ti) : i = 1, 2, . . . , n} of [f, r], we have∣∣∣∣∣

n∑
i=1

F (ti)[H(hi)−H(hi−1)]− (HKS)

∫ r

f
FdH

∣∣∣∣∣ < ε

2
· e. (7)

Similarly, there exists gauge δ2 on [r, g] such that for any δ2-fine tagged division Q =
{([ki−1, ki], si) : i = 1, 2, . . . ,m} of [r, g], we have∣∣∣∣∣

m∑
i=1

F (si)[H(ki)−H(ki−1)]− (HKS)

∫ g

r
FdH

∣∣∣∣∣ < ε

2
· e. (8)

Define a function δ : [f, g]→ C[f, g] by

δ(h) =

{ δ1(h) ∧ (r − h) , if f ≤ h ≤ r
δ1(h ∧ r) ∧ δ2(h ∨ r) , if h = r or h is incomparable to r
δ2(h) ∧ (h− r) , if r ≤ h ≤ g.

Then δ is a gauge on [f, g]. Let D = {([hi−1, hi], ti) : i = 1, 2, . . . , n} be a δ-fine tagged
division of [f, g]. By definition of δ, we have r = hi0 for some i0 ∈ {1, 2, . . . , n}. Hence,
D = D1 ∪D2 for some δ1-fine tagged division D1 of [f, r] and δ2-fine tagged division D2

of [r, g]. By (7) and (8),∣∣∣∣ n∑
i=1

F (ti)[H(hi)−H(hi−1)]−
{

(HKS)

∫ r

f
FdH + (HKS)

∫ g

r
FdH

}∣∣∣∣ < ε · e.



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 136

Therefore, F ∈ HKS([f, g], H) and

(HKS)

∫ g

f
FdH = (HKS)

∫ r

f
FdH + (HKS)

∫ g

r
FdH.

In the next theorem, we give an analogous form of Cauchy criterion for HKS−integral.

Theorem 6. (Cauchy Criterion) F ∈ HKS([f, g], H) if and only if for every ε > 0 there
exists a gauge δ on [f, g] such that for any δ-fine tagged divisions D = {([u, v], t)} and
Q = {([u′, v′], s)} of [f, g], we have∣∣∣∣∑

D

F (t)[H(v)−H(u)]−
∑
Q

F (s)[H(v′)−H(u′)]

∣∣∣∣ < ε · e.

Proof. (⇒) Let ε > 0. Then there exists a gauge δ on [f, g] such that for any δ-fine
tagged division D = {([u, v], t)} of [f, g], we have∣∣∣∣∑

D

F (t)[H(v)−H(u)]− (HKS)

∫ g

f
FdH

∣∣∣∣ < ε

2
· e. (9)

Let D = {([u, v], t)} and Q = {([u′, v′], s)} be any δ-fine tagged divisions of [f, g]. By (9)∣∣∣∣∑
D

F (t)[H(v)−H(u)]−
∑
Q

F (s)[H(v′)−H(u′)]

∣∣∣∣ < ε · e.

(⇐) By assumption, for each n ∈ N, there exists a gauge δn on [f, g] such that for any
δn-fine division D = {([u, v], t)} and Q = {([u′, v′], s)} of [f, g], we have∣∣∣∣∑

D

F (t)[H(v)−H(u)]−
∑
Q

F (s)[H(v′)−H(u′)]

∣∣∣∣ < 1

n
· e. (10)

We may assume that {δn} is decreasing; that is, δn ≥ δn+1 for all n.

Now, for each n ∈ N, fix a δn-fine tagged division Dn = {([u, v], t)} of [f, g] and we
write

rn =
∑
Dn

F (t)[H(v)−H(u)].

Note that if m ≥ n then δn ≥ δm; implying that every δm-fine tagged division of [f, g] is
also a δn-fine tagged division of [f, g]. Thus, for all m > n

|rn − rm| =
∣∣∣∣∑
Dn

F (t)[H(v)−H(u)]−
∑
Dm

F (s)[H(v′)−H(u′)]

∣∣∣∣ < 1

n
· e.



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 137

Hence, {rn} is a Cauchy sequence in C[a, b]. Since C[a, b] is complete, {rn} converges to
some r ∈ C[a, b]. We claim that

r = (HKS)

∫ g

f
FdH.

Let ε > 0. Since lim
n→∞

rn = r in C[a, b], there exists N1 ∈ N such that for any n ≥ N1,

∣∣rn − r∣∣ < e · ε
2
. (11)

By Archimedean Principle, there exists N2 ∈ N such that 1
N2

< ε
2 . Take N = N1 ∧ N2.

Define a gauge δ : [f, g] → C[a, b] by δ = δN . Let D = {([u, v], t)} be any δ-fine tagged
division of [f, g]. Note that D is also δN -fine tagged division of [f, g], N ≥ N1 and N ≥ N2.
Thus, by (10) and (11) ∣∣∣∣∑

D

F (t)[H(v)−H(u)]− r
∣∣∣∣ < ε · e.

This proves our claim.

Theorem 7. If F ∈ HKS([f, g], H) and [r, s] ⊆ [f, g], then F ∈ HKS([r, s], H).

Proof. Let ε > 0. By Theorem 6, there exists a gauge δ on [f, g] such that for any
δ-fine tagged divisions D and Q of [f, g], we have∣∣∣∣∑

D

F (t)[H(v)−H(u)]−
∑
Q

F (t)[H(v)−H(u)]

∣∣∣∣ < ε · e. (12)

Consider any δ-fine tagged divisions P1 and P2 of [r, s]. If D1 is any δ-fine tagged division
of [f, r] and D2 is any δ-fine tagged division of [s, g], then

D = D1 ∪ P1 ∪D2 and Q = D1 ∪ P2 ∪D2

are δ-fine tagged divisions of [f, g] and by (12)∣∣∣∣∑
P1

F (t)[H(v)−H(u)]−
∑
P2

F (t)[H(v)−H(u)]

∣∣∣∣ < ε · e.

By Cauchy criterion, F ∈ HKS([r, s], H).

Theorem 8. Let H : [f, g] → C[a, b] be increasing, that is, H(k) ≤ H(h) for any k ≤ h
in [f, g]. If F ∈ HKS([f, g], H) and F (h) ≥ θ for every h ∈ [f, g], then

(HKS)

∫ g

f
FdH ≥ θ.



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 138

Proof. Let ε > 0. Then there exists a gauge δ on [f, g] such that for any δ-fine tagged
division D of [f, g], we have∣∣∣∣∑

D

F (t)[H(v)−H(u)]− (HKS)

∫ g

f
FdH

∣∣∣∣ < ε · e. (13)

Since F (h) ≥ θ for all h ∈ [f, g] and H is increasing,∑
D

F (t)[H(v)−H(u)] ≥ θ.

Therefore,

θ ≤
∑
D

F (t)[H(v)−H(u)] < (HKS)

∫ g

f
FdH + ε · e.

Since ε > 0 is arbitrary,

(HKS)

∫ g

f
FdH ≥ θ.

Theorem 9. If F,G ∈ HKS([f, g], H) and F (h) ≤ G(h), for all h ∈ [f, g], then

(HKS)

∫ g

f
FdH ≤ (HKS)

∫ g

f
GdH.

Proof. Define a function E on [f, g] by setting E(h) = G(h)− F (h), for all h ∈ [f, g].
Then E(h) ≥ θ, for all h ∈ [f, g]. Since F,G ∈ HKS([f, g], H), E ∈ HKS([f, g], H) and
by Theorem 8

(HKS)

∫ g

f
EdH ≥ θ.

Hence,

θ ≤ (HKS)

∫ g

f
EdH = (HKS)

∫ g

f
(G− F )dH = (HKS)

∫ g

f
GdH − (HKS)

∫ g

f
FdH.

Therefore,

(HKS)

∫ g

f
GdH ≤ (HKS)

∫ g

f
FdH.

5. An Existence Theorem

A function F : [f, g]→ C[a, b] is bounded on [f, g] if there exists K ≥ θ in C[a, b] such
that

|F (h)| ≤ K, for all h ∈ [f, g].



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 139

A function F : [f, g] → C[a, b] is continuous at h0 ∈ [f, g], if for any ε > 0 there exists
δ = δ(h0) > θ such that whenever h ∈ [f, g] with |h− h0| < δ, we have∣∣F (h)− F (h0)

∣∣ < ε · e.

F is said to be uniformly continuous on [f, g], if for any ε > 0 there exists δ > θ such
that whenever h, h′ ∈ [f, g] with |h′ − h| < δ, we have∣∣F (h′)− F (h)

∣∣ < ε · e.

If F : [f, g]→ C[a, b] is uniformly continuous on [f, g], then it is continuous on [f, g].

Definition 3. Let D1 and D2 be tagged divisions of [f, g]. We say that D2 is finer than
D1, denoted by D1 � D2, if for every ([u, v], t) ∈ D2 there exists ([u′, v′], t′) ∈ D1 such
that [u, v] ⊆ [u′, v′], and every tag in D1 is a tag in D2. For every ([u′, v′], t′) ∈ D1,
the tagged division P = {([zi−1, zi], ti) ∈ D2 : [zi−1, zi] ⊆ [u′, v′], i = 1, 2, . . . , n} is the
refinement of ([u′, v′], t′) in D2.

We can easily see that if D1 and D2 are tagged divisions of [f, g], then there exists a
tagged division D0 of [f, g] such that D1 � D0 and D2 � D0.

Let D([f, g]) be the collection of all divisions of [f, g]. For F : [f, g] → C[a, b] and
D = {[u, v]} ∈ D([f, g]), the variation of F over D is given by

var(F,D) =
∑
D

∣∣F (v)− F (u)
∣∣.

Note that for any division D of [f, g], var(F,D) is a continuous function on [a, b]; that is,
var(F,D) ∈ C[a, b], for any D ∈ D([f, g]).

Definition 4. We say that the function F : [f, g]→ C[a, b] is of bounded variation on
[f, g] if

υF = υ(F ; [f, g]) = sup
D∈D([f,g])

var(F,D)

is continuous on [a, b]; that is, υF ∈ C[a, b].

Note that for any F : [f, g]→ C[a, b], υF is a mapping from [a, b] to [0,+∞]; that is,

0 ≤ υF (x) ≤ +∞, for all x ∈ [a, b].

Hence, if F : [f, g]→ C[a, b] is of bounded variation, then

0 ≤ υF (x) < +∞, for all x ∈ [a, b].

Theorem 10. Let H : [f, g] 7→ C[a, b] be of bounded variation. Then the variation of H
is additive; that is, if f ≤ r ≤ g, then

υ(H; [f, g]) = υ(H; [f, r]) + υ(H; [r, g]).



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 140

Proof. Suppose that H : [f, g] → C[a, b] is of bounded variation. Let r ∈ [f, g] and
D = {h0, . . . , hn} be a division of [f, g]. Then D′ = {h0, . . . , hk−1, r, hk, . . . , hn} is a
refinement of D obtained by adjoining r to D. Thus∑

D

∣∣H(v)−H(u)
∣∣ ≤∑

D1

∣∣H(v)−H(u)
∣∣+
∑
D2

∣∣H(v)−H(u)
∣∣

where D1 = {f = h0, h1, . . . , hk−1, r} and D2 = {r, hk, . . . , hn = g}. Note that D′ =
D1 ∪D2 and that∑

D1

∣∣H(v)−H(u)
∣∣ ≤ sup

D∈D([f,r])

(∑
D

∣∣H(v)−H(u)
∣∣) = υ(H; [f, r]) and

∑
D2

∣∣H(v)−H(u)
∣∣ ≤ sup

D∈D([r,g])

(∑
D

∣∣H(v)−H(u)
∣∣) = υ(H; [r, g]).

Hence,

υ(H; [f, g]) = sup
D∈D([f,g])

(∑
D

∣∣H(v)−H(u)
∣∣) ≤ υ(H; [f, r]) + υ(H; [r, g]).

On the other hand, for any D1 ∈ D([f, r]) and D2 ∈ D([r, g]), their union D′ =
D1 ∪ D2 ∈ Dr([f, g]), where Dr([f, g]) is the set of all divisions of [f, g] with r as one of
the division points. Note that Dr([f, g]) ⊆ D([f, g]). Hence,

sup
D′∈Dr([f,g])

(∑
D′

∣∣H(v)−H(u)
∣∣) ≤ sup

D∈D([f,g])

(∑
D

∣∣H(v)−H(u)
∣∣) = υ(H; [f, g])

Thus,

υ(H; [f, r]) + υ(H; [r, g]) ≤ sup
D′∈Dr([f,g])

(∑
D′

∣∣H(v)−H(u)
∣∣)

≤ υ(H; [f, g]).

Therefore, combining the two inequalities

υ(H; [f, r]) + υ(H; [r, g]) = υ(H; [f, g]).

Theorem 11. (Existence Theorem) If F : [f, g] → C[a, b] is continuous and H : [f, g] →
C[a, b] is of bounded variation on [f, g], then F ∈ HKS([f, g], H).

Proof. Let ε > 0. Since H is of bounded variation, υH ∈ C[a, b]. This means that
there exists K > 0 such that υH(x) ≤ K for all x ∈ [a, b] . Since F is continuous on
[f, g], for all h0 ∈ [f, g] there exists δ0(h0) > θ in C[a, b] such that whenever h ∈ [f, g] with
|h− h0| < δ0(h0), we have

|F (h)− F (h0)| < ε · e.



A. Cunanan, J. Benitez / Eur. J. Pure Appl. Math, 13 (1) (2020), 130-143 141

Define a gauge δ on [f, g] by δ(h) = δ0(h)
2 , for all h ∈ [f, g]. Let

D = {([f, h1], t1), ([h1, h2], t2), . . . , ([hm−1, g], tm)}

and
Q = {([f, k1], r1), ([k1, k2], r2), . . . , ([kq−1, g], rq)}

be δ-fine tagged divisions of [f, g]. Then there exists a tagged division D0 such that
D � D0 and Q � D0. Now, for every ([hi−1, hi], ti) ∈ D, f = h0, hm = g, 1 ≤ i ≤ m,
consider the difference

∆(hi−1, hi) = F (ti)
[
H(hi)−H(hi−1)

]
− S(F,H;Pi)

where

Pi =

{([
z
(i)
j−1, z

(i)
j

]
, s

(i)
j

)}ni

j−1
, z

(i)
0 = hi−1, z

(i)
ni

= hi

is the refinement of ([hi−1, hi], ti) in D0. Then

∆(hi−1, hi) =

ni∑
j=1

[
F (ti)− F (s

(i)
j )
][
H(z

(i)
j )−H(z

(i)
j−1)

]
.

Now, s
(i)
j , ti ∈ [hi−1, hi] ⊆ (ti − δ(ti), ti + δ(ti)) which implies that∣∣∣∣ti − s(i)j ∣∣∣∣ ≤ ∣∣hi − hi−1∣∣ < δ(ti).

By continuity of F at ti,∣∣∣∣s(i)j − ti∣∣∣∣ < δ(ti) =
δ0(ti)

2
< δ0(ti)⇒

∣∣F (s
(i)
j )− F (ti)

∣∣ < ε · e.

So,

|∆(hi−1, hi)| =
∣∣∣∣ ni∑
j=1

[
F (ti)− F (s

(i)
j )
][
H(z

(i)
j )−H(z

(i)
j−1)

]∣∣∣∣.
Hence, by Theorem 10, we have∣∣∣S(F,H;D)− S(F,H;D0)

∣∣∣ =

∣∣∣∣ m∑
i=1

F (ti)[H(hi)−H(hi−1)]−
m∑
i=1

S(F,H, Pi)

∣∣∣∣
=

∣∣∣∣ m∑
i=1

{
F (ti)[H(hi)−H(hi−1)]− S(F,H, Pi)

}∣∣∣∣ =

∣∣∣∣ m∑
i=1

∆(hi−1, hi)

∣∣∣∣
≤

m∑
i=1

∣∣∣∆(hi−1, hi)
∣∣∣



REFERENCES 142

=
m∑
i=1

∣∣∣∣ ni∑
j=1

[
F (ti)− F (s

(i)
j )
][
H(z

(i)
j )−H(z

(i)
j−1)

]∣∣∣∣
≤

m∑
i=1

( ni∑
j=1

∣∣F (ti)− F (s
(i)
j )
∣∣∣∣H(z

(i)
j )−H(z

(i)
j−1)

∣∣)

≤
m∑
i=1

( ni∑
j=1

ε

K
· e ·

∣∣H(z
(i)
j )−H(z

(i)
j−1)

∣∣)

≤ ε

K
· e ·

m∑
i=1

( ni∑
j=1

∣∣H(z
(i)
j )−H(z

(i)
j−1)

∣∣)

≤ ε

K
· e ·

m∑
i=1

υ(H; [hi−1, hi])

=
ε

K
· e · υH <

ε

K
· e ·K < ε · e.

By similar argument, ∣∣S(F,H;Q)− S(F,H;D0)
∣∣ < ε · e.

Thus,∣∣S(F,H;D)− S(F,H;Q)
∣∣ =

∣∣∣∣S(F,H;D)− S(F,H;D0) + S(F,H;D0)− S(F,H;Q)

∣∣∣∣
≤

∣∣∣∣S(F,H;D)− S(F,H;D0)

∣∣∣∣+

∣∣∣∣S(F,H;Q)− S(F,H;D0)

∣∣∣∣
< ε · e+ ε · e
= 2ε · e.

By Cauchy criterion, F ∈ HKS([f, g], H).

Acknowledgements

This article is funded by CHED-K12 Transition Program.

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