EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 1183-1196
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On the McShane-Dunford-Stieltjes Integral and
McShane-Pettis-Stieltjes Integral

Develin Omayan1,∗, Greig Bates Flores2

1,2 Department of Mathematics, College of Arts and Sciences, Central Mindanao University,
Philippines

Abstract. This paper combines the McShane-Stieltjes integral and Pettis approaches by utilizing
Pettis’ definition, which coincides with the Dunford integral rather than the version applicable to
weakly measurable functions with Lebesgue integrable images. In this way, another integration
process without measure theoretic standpoint will be introduced. To this end, we will define the
McShane-Dunford-Stieltjes integral and McShane-Pettis-Stieltjes integral in Banach Space and
provide its simple properties such as the uniqueness, linearity property of both the integrand
and integrator, additivity and formulate the Cauchy criterion of these integrals. In addition, the
existence theorem of McShane-Dunford Stieltjes integral will also be presented.

2020 Mathematics Subject Classifications: 40A99, 46BXX, 49Q15, 26A42, 26A45

Key Words and Phrases: McShane-Stieltjes integral, McShane-Dunford-Stieltjes integral,
McShane-Pettis-Stieltjes.

1. Introduction

The Kurzweil-Henstock integral, also known as the generalized Riemann integral, in
most cases, is an expansion of the Lebesgue integral in the context of Riemann. It is well-
known that English mathematician Ralph Henstock and Czech mathematician Jaroslav
Kurzweil provided a slight but clever alteration to the conventional Riemann integral to
obtain this definition [1]. Unlike the Lebesgue integral, which requires a foundation in
measure theory for its definition making it a difficult one, the Kurzweil-Henstock integral
is accessible through a more straightforward gauge-based approach.

Now, for real-valued functions the generalization of Riemann integral yields to an
integral referred to as the McShane integral, that is, in most cases, equivalent to the
Lebesgue integral [4]. Meanwhile, in the late 1960’s, McShane [9] proved that the Lebesgue
integral is indeed equivalent to a modified version of the Kurzweil-Henstock integral [6].
Note that the Kurzweil-Henstock and McShane integrals differ in how the tagged intervals

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5165

Email addresses: develinomayan@gmail.com (D. Omayan), f.greigbates.flores@cmu.edu.ph (G.B. Flores)

https://www.ejpam.com 1183 © 2024 EJPAM All rights reserved.



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1184

are constructed. In the Kurzweil-Henstock integral, the tags must be within the interval
while for the McShane integral the tags can be located outside the interval.

On the other hand, another integration process called Pettis integral was discovered
being an extension of the Lebesgue integral to a Banach-valued functions on a measure
space. The Pettis integral has two definitions: one for Dunford integrable functions, where
the integral coincides with the Dunford integral; the other for weakly measurable functions
with Lebesgue integrable images, ensuring the existence of an element satisfying a certain
condition for every measurable set [10]. It is also important to note that the convergence
theorems of the Dunford integral wherein in a sequence of Dunford integrable function
is uniform convergent or weakly convergent, weakly monotone and its limit exists [11].
But Gouju and Schwabik (2002) mentioned that the relation between the Pettis integral
and the McShane integral for arbitrary spaces is unknown [5]. Meanwhile, Benitez and
Flores [3] introduced the McShane-Stieltjes integral in Banach Space and provided some
of its simple properties. Furthermore, the applicability of this study spans diverse areas,
including the theory of curve integrals, probability, hysteresis, and functional-differential
and generalized differential equations.

2. Preliminaries

This section will be discussing essential notions that are necessary in formulating the
McShane-Pettis-Stieltjes integral in a Banach Space. Throughout the paper, we let X
to be a Banach Space, Rn denotes the n-Euclidean space, R+ is the set of positive real
numbers,

[a, b] =

n∏
i=1

[ai, bi],

where −∞ < ai < bi < ∞ for i = 1, · · · , n to be a compact interval in Rn , In([a, b])
is the collection of all compact subintervals of [a, b] and V ([u,v]) is the collection of all
vertices of [u,v]. Moreover, Rn is equipped with the maximum norm ∥·∥Rn . So, for each
x ∈ Rn, we define ∥·∥ the maximum norm of x by

∥x∥Rn = max{|xi| : i = 1, · · · , n},

where x = (x1, · · · , xn) and given r > 0, we set B(x, r) = {y ∈ Rn : ∥x− y∥Rn < r},
where x− y = (x1 − y2, x2 − y2, · · · , xn − yn).

Definition 1. [8] Two compact intervals [q, r], [s, t] ∈ Rn are said to be non-overlapping
if

n∏
i=1

(qi, ri)
⋂ n∏

i=1

(si, ti) = ∅,

where q = (q1, q2, · · · , qn), r = (r1, r2, · · · , rn), s = (s1, s2, · · · , sn) and t = (t1, t2, · · · , tn).

Definition 2. [8] A function δ : [a, b] → R+ is called a gauge on [a, b].



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1185

Definition 3. [10] A pair (t, [u, v]) of a point t ∈ Rn and a compact interval [u,v] of
Rn is called a tagged interval. Here, t is known as the tag of [u,v].

Definition 4. [10] A finite collection {(tk, [uk,vk]), k = 1, 2, · · · , p} of pairwise non-
overlapping tagged intervals is called an M-system in [a, b] if [uk,vk] ⊆ [a, b] for
k = 1, 2, · · · , p.

Definition 5. [10] An M -system {(tk, [uk,vk]), k = 1, 2, · · · , p} in [a, b] is called an

M-partition of the interval if

p⋃
k=1

[uk,vk] = [a, b].

Definition 6. [10] Given a gauge δ defined on {t1, · · · , tp}, a tagged interval (t, [u, v])
is said to be δ-fine if [u,v] ⊆ B(t, δ(t)), where B(t, δ(t)) is the open ball in Rn centered
at t with radius δ(t). Here, M -systems or M -partitions are called δ-fine if all the tagged
intervals (tk, [uk,vk]), where k = 1, 2, · · · , p, are δ-fine with respect to the gauge δ.

For simplicity, we denote {(tk, [uk,vk]), k = 1, 2, · · · , p} by {(t, [u,v])}.

Definition 7. [12] Let δ be a gauge defined on {t}. An M-system P = {(t, [u,v])} is
said to be a δ-fine division of [a, b] if t ∈ [u,v] ⊆ B(t, δ(t)).

Lemma 1. [8] (Cousin’s Lemma) If δ is a gauge on [a, b], then there exists a δ-fine
M-Partition of [a, b].

Definition 8. [8] Let g : [a, b] → R. The total variation of g over [a, b] is given by

V ar(g, [a, b]) = sup

{ ∑
[u,v]∈D

|∆g([u, v])| : D is a partition of [a, b]

}

where

∆g([u, v]) =
∑

t∈V ([u,v])

(
g(t)

n∏
k=1

(−1)χ{uk}(tk)

)
(⋆)

and [u, v] ∈ In
(
[a, b]

)
.

Definition 9. [8] A function g : [a, b] → R is said to be of bounded variation on [a, b]
if V ar(g, [a, b]) is finite.

Definition 10. [2] Let f be Banach-valued function defined on [a, b] and g be a real-
valued function defined [a, b]. A function f is said to be McShane-Stieltjes integrable,
or simply MS-integrable, with respect to g on [a, b] if there exists J ∈ X with the
following property: for each ε > 0 there is a gauge δ such that for every δ-fine M-partition
P = {(t, [u,v])} of [a, b] the inequality∥∥∥∥∥∥

∑
(t,[u,v])∈P

f(t)∆g([u,v])− J

∥∥∥∥∥∥
X

< ε



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1186

holds. In this case, the McShane-Stieltjes integral is J = (MS)

∫
[a,b]

fdg. For brevity,

denote S(f, g, P ) =
∑

(t,[u,v])∈P

f(t)∆g([u,v]) and MS([a, b], g) be the collection of MS-

integrable function with respect to g on [a, b].

Definition 11. [3] Let f : [a, b] → X and g : [a, b] → R be a function. We say that
f is PUL-Stieltjes integrable to A ∈ X with respect to g on [a, b] if for every ε > 0,
there exists a gauge δ on [a, b] such that for every δ-fine division D of [a, b], we have
∥S(f, g,D)−A)∥ < ε. In this case, A is the PUL-Stieltjes integral of f with respect to g

and we write A = (P )

∫
[a,b]

fdg.

Theorem 1. [2] A function f : [a, b] → X is PUL-Stieltjes integrable with respect to
g : [a, b] → R if and only if f is McShane-Stieltjes integrable with respect to g on [a, b].
Moreover,

(P )

∫
[a,b]

fdg = (MS)

∫
[a,b]

fdg.

Definition 12. [10] If f : [a, b] → X is weakly measurable such that the function
x∗(f) : [a, b] → R is McShane integrable for each x∗ ∈ X∗ then f is called Dunford

integrable. The Dunford integral (D)

∫
E
f of f over a measurable set E ⊆ [a, b] is

defined by the element x∗∗E ∈ X∗∗, that is,

(D)

∫
E
f = x∗∗E ∈ X∗∗,

where x∗∗E (x∗) = (D)

∫
E
x∗(f) for all x∗ ∈ X∗. Here, denote by D[a, b] the set of all

Dunford integrable functions on [a, b].

Definition 13. [10] If f : [a, b] → X is Dunford integrable where (D)

∫
E
(f) in X (or

more precisely (D)

∫
E
f ∈ e(X) ⊆ X∗∗, where e is the canonical embedding of X into X∗∗)

for every measurable E ⊆ [a, b], then f is called Pettis integrable and

(P )

∫
E
f = (D)

∫
E
f

is called the Pettis integral of f over the set E.

Definition 14. [7] A linear functional f is a linear operator with domain in a vector
space X and range in the scalar field K of X, that is, f : X → K, where K = R or
K = C.



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1187

Definition 15. [2] Let f : [a, b] → X be any Banach-valued function. We say that f is
continuous at a point y ∈ [a, b] if for every ε > 0, there exists a δ > 0 such that for every
x ∈ [a, b] with ∥x− y∥Rn < δ, we have |f(x)− f(y)| < ε.

Definition 16. [7] Let X be a vector space and x ∈ X. Define
X∗ = {f : X → K |f is a linear functional } and the following operations in X∗:

(i) (f1 + f2)(x) = f1(x) + f2(x); and

(ii) (αf)(x) = α(f(x)).
Then (X∗,+, ·) is a vector space called the algebraic dual space.

Definition 17. [7] Let X be a vector space and fix x ∈ X. Define
X∗∗ = {g : X∗ → K |g(f) = gx(f) = f(x) ∀ f ∈ X∗ is a linear functional } and the
following operations in X∗∗:

(i) (g1 + g2)(f) = g1(f) + g2(f); and

(ii) (αg)(f) = α(g(f)).
Then (X∗∗,+, ·) is a vector space called the second algebraic dual space.

3. McShane-Dunford-Stieltjes Integral and McShane-Pettis-Stieltjes
Integral in Banach Space

Before going through the paper an important proposition is provided.

Proposition 1. Let f : [a, b] → X and g : [a, b] → R be functions. If f is McShane-

Stieltjes integrable with (MS)

∫
[a,b]

f dg ∈ X with respect to g on [a, b]. Then for all

x∗ ∈ X∗ the real function x∗(f) : [a, b] → R is McShane-Stieltjes integrable and

(MS)

∫
[a,b]

x∗(f)dg = x∗
(
(MS)

∫
[a,b]

fdg
)
.

Proof. Suppose that f is MS-integrable to A = (MS)

∫
[a,b]

f dg with respect to g on [a, b].

Let x∗ ∈ X∗ and let ε > 0. Then there exists a gauge δ such that∥∥∥∥∥∥
∑

(t,[u,v])∈P

f(t)∆g([u,v])−A

∥∥∥∥∥∥
X

<
ε

∥x∗∥X∗ + 1
·

for every δ-fine M -partition P = {(t, [u,v])} of [a, b]. Observe that∥∥∥∥∥∥
∑

(t,[u,v])∈P

(x∗f)(t)∆g([u,v])− x∗
(
(MS)

∫
[a,b]

f dg
)∥∥∥∥∥∥

X∗



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1188

=

∥∥∥∥∥∥x∗
( ∑

(t,[u,v])∈P

f(t)∆g([u,v])

)
− x∗

(
(MS)

∫
[a,b]

f dg
)∥∥∥∥∥∥

X∗

=

∥∥∥∥∥∥x∗
( ∑

(t,[u,v])∈P

f(t)∆g([u,v])− (MS)

∫
[a,b]

f dg

)∥∥∥∥∥∥
X∗

≤ ∥x∗∥X∗

∥∥∥∥∥∥
∑

(t,[u,v])∈P

f(t)∆g([u,v])− (MS)

∫
[a,b]

f dg

∥∥∥∥∥∥
X∗

< ∥x∗∥X∗
ε

∥x∗∥X∗ + 1
= ε.

Since ε is arbitrarily chosen, we have (MS)

∫
[a,b]

x∗(f)dg = x∗
(
(MS)

∫
[a,b]

fdg
)
; which

means that x∗(f) is MS-integrable with respect to g on [a, b] integrable for all x∗ ∈ X∗.
□

In this part, the McShane-Dunford-Stieltjes integral is established as it is a necessary
concept in introducing the McShane-Pettis-Stieltjes integral.

Definition 18. Let f : [a, b] → X and g : [a, b] → R be functions. If the function
x∗(f) : [a, b] → R is McShane-Stieltjes integrable for all x∗ ∈ X∗ with respect to g on
[a, b] and if for every interval J ⊆ [a, b], there exists an element x∗∗J ∈ X∗∗ such that

x∗∗J (x∗) = (MS)

∫
J
x∗(f) dg for all x∗ ∈ X∗, then f is called McShane-Dunford-

Stieltjes integrable (or simply MDS-integrable) with respect to g on [a, b]. For an
interval J ⊆ [a, b], we write the MDS integral of f with respect to g on [a, b] by

(MDS)

∫
J
f dg = x∗∗J ∈ X∗∗.

Denote by MDS([a, b], g) the set of all McShane-Dunford-Stieltjes integrable functions
f : [a, b] → X with respect to g on [a, b].

Theorem 2. There is at most one value satisfying Definition 18.

Proof. Assume that f is MDS-integrable with respect to g on [a, b]. By Definition 18,
x∗(f) is MS- integrable with respect to g on [a, b] for all x∗ ∈ X∗. For each interval
J ⊆ [a, b], there is an element x∗∗J ∈ X∗∗ such that

x∗∗J (x∗) = (MS)

∫
J
x∗(f) dg ∀x∗ ∈ X∗.

Now, for an interval J ⊆ [a, b]. Suppose that x∗∗J , y∗∗J ∈ X∗∗ are the values of MDS
integral of f with respect to g on [a, b]. Let x∗ ∈ X∗. Then x∗(f) is McShane-Stieltjes
integrable with respect to g on [a, b]. But

x∗∗J (x∗) = (MS)

∫
J
x∗(f) dg = y∗∗J (x∗).



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1189

Hence, x∗∗J = (MDS)

∫
J
f dg = y∗∗J . This means that x∗∗J = y∗∗J . Thus, (MDS)

∫
J
f dg is

unique. □

Theorem 3. (Linearity of MDS-integral over the Integrand)
Let f1, f2 : [a, b] → X and g : [a, b] → R be functions. If f1, f2 ∈ MDS([a, b], g),

then for every α, β ∈ R, αf1 + βf2 ∈ MDS([a, b], g) and for all J ⊆ [a, b]

(MDS)

∫
J
(αf1 + βf2) dg = α · (MDS)

∫
J
f1 dg + β · (MDS)

∫
J
f2 dg.

Proof. Fix x∗ ∈ X∗. Assume that f1, f2 is MDS-integrable with respect to g on [a, b].
Then x∗(f1) and x∗(f2) is MS-integrable. This further means that x∗(f1) + x∗(f2) is
MS-integrable. Now, let α, β ∈ R. Notice that

x∗(αf1 + βf2) = x∗(αf1) + x∗(βf2)

= α · x∗(f1) + β · x∗(f2).

And so, α · x∗(f1) + β · x∗(f2) is also MS-integrable with respect to g on [a, b]. By the
linearity property of the MS-integral and for an interval J ⊆ [a, b],

(MS)

∫
J
[α · x∗(f1) + β · x∗(f2)] dg = α · (MS)

∫
J
x∗(f1)dg + β · (MS)

∫
J
x∗(f2)dg.

Fix an interval J ⊆ [a, b]. Since f1, f2 ∈ MDS([a, b], g), it follows that we can pick
operators x∗∗J , y∗∗J ∈ X∗∗ such that

x∗∗J (x∗) = (MS)

∫
J
x∗(f1) dg and y∗∗J (x∗) = (MS)

∫
J
x∗(f2) dg ∀x∗ ∈ X∗.

Now, αx∗∗J + βy∗∗J ∈ X∗∗. Take w∗∗
J = αx∗∗J + βy∗∗J . Fix x∗ ∈ X∗. Then

w∗∗
J (x∗) = (αx∗∗J + βy∗∗J )(x∗)

= αx∗∗J (x∗) + βy∗∗J (x∗)

= α(MS)

∫
J
x∗(f1) dg + β(MS)

∫
J
x∗(f2) dg

= (MS)

∫
J
α · x∗(f1) dg + (MS)

∫
J
β · x∗(f2) dg

= (MS)

∫
J
[α · x∗(f1) + β · x∗(f2)] dg

= (MS)

∫
J
x∗(αf1 + βf2) dg.

Consequently, αf1 + βf2 is MDS-integrable with respect to g on [a, b]. Therefore,

(MDS)

∫
J
(αf1 + βf2) dg = w∗∗

J = αx∗∗J + βy∗∗J

= α · (MDS)

∫
J
f1 dg + β · (MDS)

∫
J
f2 dg.

□



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1190

Theorem 4. (Linearity of MDS-integral over the Integrator)
Let f : [a, b] → X and g1, g2 : [a, b] → R be functions. If f is MDS-integrable with

respect to g1 and g2 on [a, b] , then f ∈ MDS([a, b], αg1 + βg2) and for every J ⊆ [a, b]

(MDS)

∫
J
f d[αg1 + βg2] = α · (MDS)

∫
J
f dg1 + β · (MDS)

∫
J
f dg2.

Proof. Fix x∗ ∈ X∗. Assume that f is McShane-Dunford-Stieltjes integrable with respect
to g1 and g2 on [a, b]. Then x∗(f) is MS-integrable with respect to g1 and g2 on [a, b].
Utilizing the linearity property of MS-integral over an integrator and for an interval
J ⊆ [a, b],

(MS)

∫
[a,b]

x∗(f)d[αg1 + βg2] = α(MS)

∫
[a,b]

x∗(f)dg1 + β(MS)

∫
[a,b]

x∗(f)dg2.

Let an interval J ⊆ [a, b]. Since f is MDS-integrable with respect to g1 and g2 on [a, b],
it implies that we can choose operators x∗∗J , y∗∗J ∈ X∗∗ such that

x∗∗J (x∗) = (MS)

∫
J
x∗(f) dg1 and y∗∗J (x∗) = (MS)

∫
J
x∗(f) dg2 ∀x∗ ∈ X∗.

Note that αx∗∗J + βy∗∗J ∈ X∗∗. Then write w∗∗
J = αx∗∗J + βy∗∗J . Fix x∗ ∈ X∗. Now,

w∗∗
J (x∗) = (αx∗∗J + βy∗∗J )(x∗)

= αx∗∗J (x∗) + βy∗∗J (x∗)

= α(MS)

∫
J
x∗(f) dg1 + β(MS)

∫
J
x∗(f) dg2

= (MS)

∫
J
x∗(f)d[αg1 + βg2].

And so, we see that f is MDS-integrable with respect to αg1 + βg2. Thus,

(MDS)

∫
J
f d[αg1 + βg2] = w∗∗

J = αx∗∗J + βy∗∗J

= α · (MDS)

∫
J
f dg1 + β · (MDS)

∫
J
f dg2. □

Theorem 5. Let g : [a, b] → R be a function. A function f : [a, b] → X is McShane-
Dunford-Stieltjes integrable with respect to g on [a, b] if and only if x∗(f) : [a, b] → R is
McShane-Stieltjes integrable with respect to g on [a, b] for all x∗ ∈ X∗.

Proof. Suppose that f is MDS-integrable with respect to g on [a, b]. By Definition 18,
x∗(f) is McShane-Stieltjes integrable with respect to g on [a, b] and so we are done.
Conversely, assume that x∗(f) : [a, b] → R is McShane-Stieltjes integrable with respect
to g on [a, b] for all x∗ ∈ X∗. Let an interval J ⊆ [a, b]. Then x∗(f) is McShane-

Stieltjes integrable with respect to g on J implying that (MS)

∫
J
x∗(f)dg ∈ R. Now, let



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1191

x∗∗ : X∗ → R by setting x∗∗J (x∗) = (MS)

∫
J
x∗(f)dg for all x∗ ∈ X∗. This means that

x∗∗J ∈ X∗∗. Hence, f is MDS-integrable with respect to g on [a, b]. □

Theorem 6. (Cauchy Criterion of the MDS-integral)
Let f : [a, b] → X and g : [a, b] → R be functions. A function f is McShane-Dunford-

Stieltjes integrable with respect to g on [a, b] if and only if for every ε > 0, there exists a
gauge δ : [a, b] → R+ such that if N1 and N2 are two δ-fine M -partitions, then

∥S(f, g,N1)− S(f, g,N2)∥X < ε.

Proof. Suppose that f is McShane-Dunford-Stieltjes integrable with respect to g on [a, b].
By Theorem 5, x∗(f) : [a, b] → R is McShane-Stieltjes integrable with respect to to g on
[a, b] for all x∗ ∈ X∗. If x∗ ∈ X∗ is the zero map so that x∗(x) = 0 for all x ∈ X. Then we
can choose δ : [a, b] → R+ such that for any two δ-fine M -partitions N1 and N2 of [a, b],

∥S(x∗(f), g,N1)− S(x∗(f), g,N2)∥X∗ = ∥x∗S(f, g,N1)− x∗S(f, g,N2)∥X∗ = ∥0− 0∥ < ε.

If x∗ ∈ X∗ is not the zero map. Then we can pick δ : [a, b] → R+ such that for every two
δ-fine M -partitions N1 and N2 of [a, b], we have

∥x∗[S(f, g,N1)− S(f, g,N2)]∥X∗ = ∥S(x∗(f), g,N1)− S(x∗(f), g,N2)∥X∗ < ∥x∗∥X∗ · ε.

This implies that

∥x∗∥X∗∥S(f, g,N1)− S(f, g,N2)∥X∗ < ∥x∗∥X∗ · ε.

In other words,
∥S(f, g,N1)− S(f, g,N2)∥X < ε.

Conversely, let x∗ ∈ X∗. Suppose that for every ε > 0, there exists a gauge δ : [a, b] → R+

such that if N1 and N2 are two δ-fine M -partitions, then

∥S(f, g,N1)− S(f, g,N2)∥X <
ε

∥x∗∥X∗ + 1
(1)

By (1), we obtain

∥S(x∗(f), g,N1)− S(x∗(f), g,N2)∥X∗ = ∥x∗[S(f, g,N1)− S(f, g,N2)]∥X
≤ ∥x∗∥X∗∥S(f, g,N1)− S(f, g,N2)∥X
< ∥x∗∥X∗ ·

ε

∥x∗∥X∗ + 1

< ε.

This implies that x∗(f) : [a, b] → R is MS-integrable with respect to g on [a, b]. Fix
J ⊆ [a, b] be a compact interval. Define x∗∗ : X∗ → R such that

x∗∗J (x∗) = (MS)

∫
J
x∗(f)dg.



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1192

Now, by definition of X∗∗, x∗∗J ∈ X∗∗ which means that x∗∗J is linear. Consequently,

x∗∗J = (MDS)

∫
J
fdg. Thus, f is MDS-integrable with respect to g on [a, b]. □

Theorem 7. (Additivity of the MDS-integral)
Let f : [a, b] → X, g : [a, b] → R and I,J ∈ In

(
[a, b]

)
that forms an M -partition of

[a, b]. Assume that f ∈ MDS(I, g) ∩MDS(J , g), then f ∈ MDS([a, b], g) and

(MDS)

∫
[a,b]

fdg = (MDS)

∫
I
fdg + (MDS)

∫
J
fdg.

Theorem 8. Let f : [a, b] → X and g : [a, b] → R be functions. If f is MDS-integrable
with respect to g on [a, b, then for each J ⊆ In

(
[a, b]

)
,

(MDS)

∫
J
fdg = (MDS)

∫
[a,b]

f · χJdg.

Proof. Let f, g : [a, b] → X be functions and J ∈ In
(
[a, b]

)
. Assume that f is MDS-

integrable with respect to g over [a, b]. If J = [a, b], then

(MDS)

∫
J
f · χJ dg = (MDS)

∫
[a,b]

f · χ[a,b] dg = (MDS)

∫
[a,b]

f dg = (MDS)

∫
J
f dg.

If I ⊂ [a, b], then f is MDS-integrable with respect to g on J . Notice that

[a, b] = ([a, b]∖ J) ∪ J .

Now,

(MDS)

∫
[a,b]

f · χJ dg = (MDS)

∫
[a,b]∖J

f · χJ dg + (MDS)

∫
J
f · χJ dg

= (MDS)

∫
[a,b]∖J

f · (0) dg + (MDS)

∫
J
f · (1) dg

= (MDS)

∫
J
f dg.

Thus,

(MDS)

∫
[a,b]

f · χJ dg = (MDS)

∫
J
f dg. □

.

Proposition 2. If f : [a, b] → X is continuous on [a, b], then for each x∗ ∈ X∗,
x∗(f) : [a, b] → R is also continuous on [a, b].

Proof. Let ε > 0 and x∗ ∈ X∗. Suppose that f : [a, b] → X is continuous on [a, b]. Then
for every ε > 0, there exists a δ > 0 such that for every u,v ∈ [a, b] with ∥u− v∥Rn < δ,
we have

|f(u)− f(v)| < ε

∥x∗∥X∗ + 1
.



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1193

Now,

∥x∗(f(u))− x∗(f(v))∥X∗ = ∥x∗[f(u)− f(v)]∥X∗

≤ ∥x∗∥X∗∥f(u)− f(v)∥
≤ ∥x∗∥X∗ |f(u)− f(v)|

< ∥x∗∥X∗
ε

∥x∗∥X∗ + 1
= ε.

And so, x∗(f) is continuous. □

Theorem 9. (Existence Theorem of MDS-integral)
Let f be continuous on [a, b] and g be a function of bounded variation. Then f is

MDS-integrable with respect to g on [a, b].

Proof. Fix x∗ ∈ X∗. Suppose that f is continuous on [a, b] and g be a function of bounded
variation. Applying Proposition 2, x∗(f) is continuous. By the Existence Theorem of
the PUL-Stieltjes integral and Theorem 1, x∗(f) is continuous and g is a function of
bounded variation means that the McShane-Stieltjes integral of x∗(f) with respect to
g on [a, b] exists. And so, x∗(f) is MS-integrable with respect to g on [a, b]. Let J
be a compact subinterval of [a, b]. Then x∗(f) is also MS-integrable with respect to

g on J . Take x∗∗J (x∗) = (MS)

∫
J
x∗(f)dg implying that x∗∗J ∈ X∗∗. This means that

x∗∗J = (MDS)

∫
J
fdg. Therefore, f is MDS-integrable with respect to g on [a, b]. □

Definition 19. Let f : [a, b] → X and g : [a, b] → R be functions. If f is MDS-

integrable with respect to g on [a, b] where (MDS)

∫
J
fdg ∈ X for every interval J ⊆ [a, b]

(more precisely (MDS)

∫
J
fdg ∈ e(X) ⊆ X∗∗, where e is the canonical embedding of X

into X∗∗), then f is called McShane-Pettis-Stieltjes integrable (or simply MPS-
integrable) with respect to g on [a, b] and

(MDS)

∫
J
f dg = (MPS)

∫
J
f dg ∈ X

is the McShane-Pettis-Stieltjes integral of f with respect to g on J ⊆ [a, b]. Here,
we write MPS([a, b], g) the set of all McShane-Pettis-Stieltjes integrable functions
f : [a, b] → X with respect to g on J ⊆ [a, b].

Theorem 10. There is at most one value satisfying Definition 19

Proof. Suppose that (MPS)

∫
J
f dg exists ∀ J ⊆ [a, b]. By Definition 19,

(MDS)

∫
J
f dg = (MPS)

∫
J
f dg.



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1194

Now, fix J ⊆ [a, b]. From Theorem 2, (MDS)

∫
J
f dg is unique, then (MPS)

∫
J
f dg is

also unique. □

Theorem 11. (Linearity of Integrand of the MPS-integral)
Let f1, f2 : [a, b] → X and g : [a, b] → R be functions. If f1, f2 ∈ MPS([a, b], g), then for
every α, β ∈ R, αf1 + βf2 ∈ MPS([a, b], g) and for all J ⊆ [a, b]

(MPS)

∫
J
(αf1 + βf2) dg = α · (MPS)

∫
J
f1 dg + β · (MPS)

∫
J
f2 dg.

Proof. Suppose that f1, f2 are MPS-integrable with respect to g on [a, b]. By Definition
19, f1, f2 are MDS-integrable with respect to g on [a, b]. Now, let α, β ∈ R and J ⊆ [a, b].
By linearity property of functions, α · f1 + β · f2 is also MDS-integrable with respect to
g on [a, b] and

(MDS)

∫
J
(α · f1 + β · f2) dg = α · (MDS)

∫
J
f1 dg + β · (MDS)

∫
J
f2 dg. (2)

By the definition of MPS-integral,

(MDS)

∫
J
f1 dg, (MDS)

∫
J
f2 dg ∈ e(X).

Set m1,m2 ∈ X so that

(MDS)

∫
J
f1 dg = m1 and (MDS)

∫
J
f2 dg = m2.

So we have α ·m1 + β ·m2 ∈ X. By equation (2),

(MDS)

∫
J
(α · f1 + β · f2) dg = α · (MDS)

∫
J
f1 dg + β · (MDS)

∫
J
f2 dg

= α · e(m1) + β · e(m2)

= e(αm1 + βm2) ∈ e(X).

This implies that α · f1 + β · f2 is MPS-integrable with respect to g on [a, b] and

(MPS)

∫
J
(α · f1 + β · f2) dg = (MDS)

∫
J
(α · f1 + β · f2) dg

= α · (MDS)

∫
J
f1 dg + β · (MDS)

∫
J
f2 dg

= α · (MPS)

∫
J
f1 dg + β · (MPS)

∫
J
f2 dg.

Hence, the proof. □



D. Omayan, G.B. Flores / Eur. J. Pure Appl. Math, 17 (2) (2024), 1183-1196 1195

Theorem 12. (Linearity of Integrator of the MPS-integral)
Let f : [a, b] → X and g1, g2 : [a, b] → R be functions. If f is MPS-integrable with

respect to g1 and g2 on [a, b], then for any α, β ∈ R, f ∈ MPS([a, b], αg1 + βg2) and

(MPS)

∫
J
f d[αg1 + βg2] = α · (MPS)

∫
J
f dg1 + β · (MPS)

∫
J
f dg2.

for every J ⊆ [a, b].

Proof. The proof is analogous to Theorem (11). □

Theorem 13. (Cauchy Criterion of the MPS-integral)
Let f : [a, b] → X and g : [a, b] → R be functions. Then f is McShane-Pettis-Stieltjes

integrable with respect to g on [a, b] if and only if for every ε > 0, there exists a gauge
δ : [a, b] → R+ such that if M1 and M2 are two δ-fine M -partitions, then

∥S(f, g,M1)− S(f, g,M2)∥X < ε.

Proof. Suppose that f is MPS-integrable with respect to g on [a, b]. By Definition 19, f
is MDS-integrable with respect to g on [a, b]. Using the Cauchy Criterion of the MDS-
integral, there exists a gauge
δ : [a, b] → R+ such that if M1 and M2 are two δ-fine M -partitions, then

∥S(f, g,M1)− S(f, g,M2)∥X < ε.

Conversely, assume that for every ε > 0, there exists a gauge δ : [a, b] → R+ such that if
M1 and M2 are two δ-fine M -partitions, then

∥S(f, g,M1)− S(f, g,M2)∥e(X) < ε.

By Cauchy Criterion on MDS-integral, x∗(f) : [a, b] → R is MS-integrable. Let an

interval J ⊆ [a, b]. Define x∗∗ : X∗ → R such that x∗∗J (x∗) = (MS)

∫
[a,b]

x∗(f)dg. By

definition of X∗∗, x∗∗J ∈ X∗∗. This means that x∗∗J = (MDS)

∫
[a,b]

fdg ∈ e(X). Hence,

(MDS)

∫
[a,b]

fdg = (MPS)

∫
[a,b]

fdg. Therefore, f is MPS-integrable with respect to g

on [a, b]. □

Theorem 14. (Additivity of the MPS-integral)
Let f : [a, b] → X and g : [a, b] → R be functions and I,J ∈ In

(
[a, b]

)
that forms an

M -partition of [a, b]. If f ∈ MPS(I, g) ∩MPS(J , g), then f ∈ MPS([a, b], g) and

(MPS)

∫
[a,b]

fdg = (MPS)

∫
I
fdg + (MPS)

∫
J
fdg.



REFERENCES 1196

Acknowledgements

The authors express their gratitude to the Department of Science and Technology-
Science Education Intitute- Science and Technology Regional Alliance of inclusive Uni-
versities for National Development (DOST-SEI-STRAND) for their financial support in
conducting this research.

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