EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

2025, Vol. 18, Issue 2, Article Number 5611

ISSN 1307-5543 – ejpam.com

Published by New York Business Global

The Saks-Henstock Lemma and the Change of Variable

Formula of the PU integral

Greig Bates C. Flores1,∗, Ann Leslie V. Flores1

1 Department of Mathematics, Central Mindanao University, Maramag, Bukidnon,

Philippines

Abstract. Henstock integral is a generalized version of the Riemann integral and in most cases,

it is more general than the Lebesgue integral that is not constructed through a measure theoretic

standpoint. The PU integral, on one hand, is a Henstock type that utilizes the notion of a partition

of unity. In this paper, the Saks-Henstock Lemma and the Change of variable formula for the PU

integral will be established.

2020 Mathematics Subject Classifications: 28B05, 26A39, 32C09

Key Words and Phrases: Gauge integrals, Partition of Unity, Change of Variable Formula,

Saks-Henstock

1. Introduction

Henstock integral is an integration process that is anchored on how the Riemann

integral is constructed. More precisely, in most cases, it is more general than the Lebesgue

integral and its construction is free from a measure theoretic standpoint. Thus, it is,

relatively, easier than how the Lebesgue integral was constructed. Its definition is through

a covering system called δ-fine, where δ is a postive function. A number of its variants were

established and one of those is the PU integral, a gauge type of definition that is anchored

in the perspective of a covering systems though partitions of unity. In fact, [1] mentioned

that the PU integral can be used in the integration of functions on manifolds. In [2, 3],

a Henstock-Kurzweil integral type were established and their application to functions on

manifolds were presented.

A finite collection of point-interval pair {(ti, Ii)}mi=1, where Ii is a compact interval, is

of Perron type if ti ∈ Ii for all i ≤ m. For a subset E of Rn, the support of a real-valued

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5611

Email addresses: greigbates.flores@cmu.edu.ph (G. Flores),

f.annleslie.flores@cmu.edu.ph (A. Flores)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 2 of 16

function f on E, written as suppf is the closure of the set of points in the domain whose

image under f is nonzero. Moreover, given a gauge δ on [a, b], a finite collection of point

interval pairs, of Perron type, {(ti, Ii)}mi=1 is said to be δ-fine Perron partition of [a, b] if

Ii is a partition of [a, b] and Ii ⊆ B(ti, δ(ti)). If for each i ≤ m, the condition ti ∈ Ii
is removed, then we obtain the McShane integral. On another note, a partition of unity

[1, 2] on a compact interval E ⊆ Rn is a finite collection {φi}mi=1 smooth functions with

the following conditions:

(i) φi ≥ 0 on E;

(ii)
m∑
k=1

φk = 1 a.e. on E.

If

m∑
k=1

φk ≤ 1 a.e. on E, then {φi}mi=1 is said to be a partial partition of unity. A

finite collection of triples {(ξi, Ii, φ)}mi=1 is said to be δ-fine PU-division of E if for each

i ≤ m, supp φi ⊆ Ii and Ii ⊆ B(ξi, δ(ξi)). Boonpogkrong, revisited the notion of the PUL

integral and it’s application in the integrals of a function defined on a manifold. Moreover,

Flores and Benitez [4, 5] generalizes the notion in its Stieltjes form and establsihed some

convergence theorems. In this paper, the definition of the PU integral will be discussed

and the Change of variable formula and the Saks-Henstock Lemma of this integral will be

established.

2. Preliminaries

In this section, we introduce the PU integral of a function defined on a Manifold.

Throughout the rest of this chapter, if no confusion arises, we denote M as a manifold.

Definition 1. [3, 4] Let X be a Banach space and let f : [a, b] → X . We define the PU

sum by

S(f,D) =
m∑
k=1

f(ξk) (R)

∫
Ik

φk

where D is a δ-fine division of [a, b] and (R)
∫
Ik
φk is the Riemann integral of φk for all

k ∈ {1, 2, · · ·m}. For brevity, we write a δ-fine division of [a, b] by D = {(ξ, I, ψ)} and a

PU sum of f over D by

S(f,D) = (D)
∑

f(ξ)

∫
I
ψ =

∑
D

f(ξ)

∫
I
ψ .



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 3 of 16

Definition 2. [4] Let X be a Banach space and f : [a, b] → X be a Banach-valued

funtion. We say that f is PU integrable to a vector A over [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,D)−A∥ < ϵ.

If A is the PU integral of f with over [a, b], then we write

A = (P)

∫
[a,b]

f.

Definition 3. [6] Let g : [a, b] → R. The total variation of g over [a, b] is given by

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

{ ∑
[u,v]∈D

∣∣∆g([u, v])
∣∣ : D is a division of [a, b]

}
,

where

∆g([u, v]) =
∑

t∈V[u,v]

g(t)
n∏
k=1

(−1)χ{uk}(tk), t = (t1, t2, · · · , tn)

and V[u, v] is the set of vertices in [u, v] =
∏n
k=1[uk, vk]. If V (g, [a, b]) < +∞, then g is

said to be a function of bounded variation on [a, b].

Example 1.

(i) For n = 2, we have

∆g([u1, v1])× [u2, v2] = g(v1, v2)− g(u1, v2) + g(u1, u2)− g(v1, u2);

(ii) For n = 1, we have

∆g([u1, v1]) = g(v1)− g(u1).

Remark 1. The total variation in n-dimensional Euclidean space is an extension of the

usual total variation in Euclidean space.

Definition 4. [7] A topological space is second countable if it has a countable basis.

Definition 5. [7] A topological spaceM is locally Euclidean of dimension n if every point

p ∈M has a neighborhood Up such that there is a homeomorphism ϕ from Up onto an open

subset Op of Rn. We call the pair (U, ϕ : U → Rn) a chart, U a coordinate neighborhood

or a coordinate open set, and ϕ a coordinate map or coordinate system on U . We say that

a chart (U, ϕ) is centered at p ∈ U if ϕ(p) = 0. A chart (U, ϕ) about p simply means that

(U, ϕ) is a chart and p ∈ U .

Definition 6. [7] A topological manifold of dimension n is a Hausdorff, second countable,

locally Euclidean space of dimension n.

Example 2. [7] The Euclidean space Rn is covered by a single chart (Rn, 1Rn), where

1Rn : Rn → Rn is the identity map. It is a prime example of a topological manifold. Every

open subset U of Rn is also a topological manifold, with chart (U , 1Rn).



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 4 of 16

3. Main Results

Lemma 1. Let U be an open subset of a compact interval E∗ in Rr and ψ : U → ψ(U)

be C1-diffeomorphism which is monotone. Suppose that E is a compact interval such that

E ⊆ ψ(U) ⊆ Rn and φ : E → R is continuous and g : E → R be a function of bounded

variation. Then

(R)

∫
E
φ dg = (R)

∫
ψ−1(E)

(φ ◦ ψ)|detψ| dg,

where (R)

∫
E
φ dg is the Riemann-Stieltjes integral of φ with respect to g on E and |detψ|

is the Euclidean norm of the partial derivatives of ψ.

Proof : Since φ is continuous and g is of bounded variation, then (R)

∫
E
φ dg exists in

R. Thus, for each ϵ > 0, there exists a constant δ1 > 0 such that for any δ1-fine division

D = {(ξ, I)} of E, we have∣∣∣∣∑
D

φ(ξ)∆g(I)− (R)

∫
E
φ dg

∣∣∣∣ < ϵ

2
. (3.1)

Note that, (φ ◦ ψ)|detψ| is continuous on ψ−1(E). Also, since g is a function of bounded

variation on ψ−1(E), it follows that

(R)

∫
ψ−1(E)

(φ ◦ ψ)|detψ| dg

exists in R; hence there exists a constant δ0 ≤ δ1 such that for any δ0-fine division D′ =

{(t,K)} of ψ−1(E), we have∣∣∣∣∑
D′

(φ ◦ ψ)(t)∆(g◦ψ)(K)− (R)

∫
ψ−1(E)

(φ ◦ ψ)|detψ| dg
∣∣∣∣ < ϵ

2
. (3.2)

Since ψ is a diffeomorphism on U , ψ is continuous at every point in U . Since E ⊆ ψ(U),

we have ψ−1(E) ⊆ U . Hence, ψ is continuous at every point in ψ−1(E). Thus, for each

x ∈ ψ−1(E) and δ1 > 0 there exists δ2(x) > 0 such that for any y ∈ B(x, δ2(x)),

|ψ(y)− ψ(x)| < δ1.

For each x ∈ ψ−1(E), let δ(x) = min{δ0(x), δ2(x)}. Now, let D0 = {(η,J)} be a fix δ-fine

division of ψ−1(E). Hence, for each (η,J) in D0 and x ∈ J we have

|ψ(η)− ψ(x)| < δ1. (3.3)

Also, ψ(J) is a compact interval in Rn that partitions E. By (3.3), {(ψ(η), ψ(J))} is

δ1-fine division of E. Thus, by (3.1)

ϵ

2
>

∣∣∣∣(R)

∫
E
φ dg −

∑
D0

φ(ψ(η)) ·∆g(ψ(J))

∣∣∣∣



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 5 of 16

=

∣∣∣∣(R)

∫
E
φ dg −

∑
D0

(φ ◦ ψ)(η) ·∆g◦ψ(J)

∣∣∣∣ (3.4)

Note that since δ ≤ δ0, D0 is also a δ0-fine division of ψ−1(E). Hence, by (3.2)∣∣∣∣∑
D0

(φ ◦ ψ)(η)∆g◦ψ(J)− (R)

∫
ψ−1(E)

(φ ◦ ψ)| detψ| dg
∣∣∣∣ < ϵ

2
. (3.5)

Therefore, by (3.4) and (3.5)

0 ≤
∣∣∣∣(R)

∫
E
φ dg − (R)

∫
ψ−1(E)

(φ ◦ ψ)|detψ| dg
∣∣∣∣

=

∣∣∣∣(R)

∫
E
φ dg −

∑
D0

(φ ◦ ψ)(t)∆(g◦ψ)(J)

+
∑
D0

(φ ◦ ψ)(t)∆(g◦ψ)(J)− (R)

∫
ψ−1(E)

(φ ◦ ψ)| detψ| dg
∣∣∣∣

≤
∣∣∣∣(R)

∫
E
φ dg −

∑
D0

(φ ◦ ψ)(t)∆(g◦ψ)(J)

∣∣∣∣
+

∣∣∣∣∑
D0

(φ ◦ ψ)(t)∆(g◦ψ)(J)− (R)

∫
ψ−1(E)

(φ ◦ ψ)|detψ| dg
∣∣∣∣

<
ϵ

2
+
ϵ

2

= ϵ.

Since ϵ > 0 is arbitrary, we have

(R)

∫
E
φ dg = (R)

∫
ψ−1(E)

(φ ◦ ψ)|detψ| dg. □

Theorem 1. (Change of Variable Formula). Let f : [a, b] → X be PU integrable

over [a, b]. Let U be an open subset of a compact interval E in Rm. Let Ψ : U → Ψ(U)

be C1-diffeomorphism and [a, b] ⊆ Ψ(U) ⊆ Rn. Then (f ◦ Ψ) · |detψ| · χΨ−1([a,b]) is PU

integrable over E and

(P)

∫
E
(f ◦Ψ) · | detΨ| · χΨ−1([a,b]) = (P)

∫
Ψ−1([a,b])

(f ◦Ψ) · | detΨ| = (P)

∫
[a,b]

f.

Proof : Fix ϵ > 0. Since f is PU integrable over [a, b], we choose a gauge δ0 on [a, b] such

that ∥∥∥∥∑
D

f(ξ)

∫
I
φ− (P)

∫
[a,b]

f

∥∥∥∥ < ϵ

2



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 6 of 16

for every δ0-fine division D = {(ξ, φ, I)} of [a, b]. Since Ψ is a C1-diffeomorphism and

U is an open set in Rr, Ψ(U) is open in Rn. So, choose δ0 in such a way that for each

ξ ∈ Ψ(U),

B(ξ, δ0(ξ)) ⊆ Ψ(U).

Define δ : E → R+ such that for u ∈ Ψ−1([a, b]), we have

B(u, δ(u)) ⊆ Ψ−1

(
B

(
Ψ(u),

δ0(Ψ(u))

2
√
p

))
, (3.6)

where 1 ≤ p is a fixed positive real number; also, for u ∈ E \Ψ−1([a, b]),

B(u, δ(u)) ∩E ⊆ E ∖Ψ−1([a, b]). (3.7)

Let D = {(ξ, I, φ)} be a δ-fine division of E. Suppose D = D1 ∪ D2 where D1 =

{(ξ, I, φ) ∈ D : ξ ∈ Ψ−1([a, b])} and D2 = D ∖D1. We may assume that

D1 = {(ξk, Ik, φk)}rk=1 and D2 = {(ξk, Ik, φk)}sk=r+1.

Let k ∈ {r + 1, r + 2, · · · , s} and let x ∈ supp φk. Here, φk(x) > 0. Since D is a δ-fine

division of E,

supp φk ⊆ Ik ⊆ B(ξk, δ(ξk));

Which implies

supp φk = supp φk ∩E ⊆ B(ξk, δ(ξk)) ∩E.

And by (3.7),

supp φk ⊆ B(ξk, δ(ξk)) ∩E ⊆ E \Ψ−1([a, b]);

which means supp φk ∩ Ψ−1([a, b]) = ∅. Thus, for each x ∈ Ψ−1([a, b]), φk(x) = 0 for

all k = r + 1, r + 2 · · · , s. Now, for each x ∈ Ψ−1([a, b])

s∑
k=r+1

φk(x) = 0

and so
s∑

k=1

φk(x) =

r∑
k=1

φk(x) +

s∑
k=r+1

φk(x) =

r∑
k=1

φk(x).

For each k = 1, 2, · · · r, let xk = Ψ(ξk) and σk = φk ◦Ψ−1.

Note that σj : Ψ(U) → R and

B

(
σk(Ψ(y)),

δ0(Ψ(y))

2
√
p

)
⊆ B

(
σk(Ψ(y)), δ0(Ψ(y))

)
.



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 7 of 16

Notice that x ∈ supp σk · χ[a,b] for all k = 1, 2, · · · , r; Then

φk(Ψ
−1(x)) = (φk ◦Ψ−1)(x) = σk(x) > 0,

that is, Ψ−1(x) ∈ supp φk. From (3.6) and since D is a δ-fine division of E, we have

Ψ−1(x) ∈ supp φk ⊆ Ik ⊆ B(ξk, δ(ξk)) ⊆ Ψ−1

(
B

(
Ψ(ξk),

δ0(Ψ(ξk))

2
√
p

))
.

Hence,

x ∈ B

(
Ψ(ξk),

δ0(Ψ(ξk))

2
√
p

)
.

Thus, for each k = 1, 2, · · · , r,

supp σk · χ[a,b] ⊆ (B

(
Ψ(ξk),

δ0(Ψ(ξk))

2
√
p

))
∩ [a, b].

Now, since B

(
σk(Ψ(y)), δ0(Ψ(y))

2
√
p

)
⊆ B

(
σk(Ψ(y)), δ0(Ψ(y))

)
for all k = 1, 2, · · · r, it

follows that

supp σk · χ[a,b] ⊆ B

(
σk(Ψ(y)),

δ0(Ψ(y))

2
√
p

)
∩ [a, b]

⊆ B

(
σk(Ψ(y)), δ0(Ψ(y))

)
∩ [a, b]

for all k = 1, 2, · · · r. So, we choose a compact interval Jk such that

B

(
σk(Ψ(y)),

δ0(Ψ(y))

2
√
p

)
∩ [a, b] ⊆ Jk ⊆ B

(
σk(Ψ(y)), δ0(Ψ(y))

)
∩ [a, b].

for all k = 1, 2, · · · , r. Thus,

supp σk · χ[a,b] ⊆ B

(
σk(Ψ(y)),

δ0(Ψ(y))

2
√
p

)
∩ [a, b] ⊆ Jk

⊆ B

(
σk(Ψ(y)), δ0(Ψ(y))

)
∩ [a, b],

that is,

supp σk · χ[a,b] ⊆ Jk ⊆ B

(
σk(Ψ(y)), δ0(Ψ(y))

)
∩ [a, b] (3.8)

for all k = 1, 2, · · · , r. Note that on [a, b], χ[a,b] = 1. Since φk and Ψ−1 are continu-

ously differentiable functions on [a, b], σk · χ[a,b] = σk = φk ◦ Ψ−1 is also continuously

differentiable function on [a, b]. Fix x ∈ [a, b]. Since φk is a partition of unity for all

k = 1, 2, · · · , r, we have

r∑
k=1

σk · χ[a,b](x) =

r∑
k=1

σk(x) =

r∑
k=1

(φk(Ψ
−1(x)) = 1.



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 8 of 16

This means that {σk · χ−1} is a partition of unity on [a, b]. The inclusion (3.8) implies

that D0 = {(xk,Jk, σk · χ−1)}rk=1 is a δ0-fine division of [a, b].

Next, we will show that for each k = 1, 2, · · · , r, supp σk ◦ Ψ = supp φk. Indeed, let

k ∈ {1, 2, · · · , r} and let x ∈ supp σk ◦ Ψ. Then σk(Ψ(x)) > 0 and so Ψ(x) ∈ supp σk =

supp φk ◦Ψ−1. Hence, x ∈ supp φk and supp σk ◦Ψ ⊆ supp φk. Now, let k ∈ {1, 2, · · · , r}
and x ∈ supp φk. Then φk(Ψ

−1(Ψ(x))) = φk(x) > 0; and so Ψ(x) ∈ supp φk ◦ Ψ−1 =

supp σk and σk(Ψ(x)) = (σk ◦ Ψ)(x) > 0. Henceforth, x ∈ supp σk ◦ Ψ and supp φk ⊆
supp σk ◦Ψ. Thus, supp φk = supp σk ◦Ψ. By Lemma 1,∫

Jk

σk =

∫
Ψ−1(Jk)

(σk ◦ ψ)|detΨ| =
∫
Ik

φk| detΨ|.

Notice that∥∥∥∥ s∑
k=1

f(Ψ(ξk))|detΨ|χΨ−1([a,b])(ξk)

∫
Ik

φk −
p∑

k=1

f(xk)

∫
Jk

σkχ[a,b]

∥∥∥∥
=

∥∥∥∥ p∑
k=1

f(Ψ(ξk))|detΨ|
∫
Ik

φk −
p∑

k=1

f(xk)

∫
Jk

σk

∥∥∥∥
=

∥∥∥∥ p∑
k=1

f(Ψ(ξk))|detΨ|
∫
Ik

φk −
p∑

k=1

f(Ψ(ξk))|detΨ|
∫
Ik

φk

∥∥∥∥
= 0

implies
s∑

k=1

f(Ψ(ξk))| detΨ|χΨ−1(E)(ξk)

∫
Ik

φk =

p∑
k=1

f(xk)

∫
Jk

σkχE . (3.9)

Since D0 is a δ0-fine division of [a, b],∥∥∥∥ p∑
k=1

f(xk)

∫
Jk

σkχE − (P)

∫
[a,b]

f

∥∥∥∥ < ϵ;

that is, ∥∥∥∥ s∑
k=1

f(Ψ(ξk))|detΨ|χΨ−1(E)(ξk)

∫
Ik

φk − (P)

∫
[a,b]

f

∥∥∥∥ < ϵ

This simply means, (f ◦Ψ)χΨ−1([a,b]) is PU integrable with over E and

(P)

∫
[a,b]

f = (P)

∫
Ψ−1([a,b])

(f ◦Ψ) dΨ. □



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 9 of 16

Lemma 2. Let Dp = {ξ, I, φ} be a partial δ-fine division of [a, b], σ(x) =
∑

D1
φ(x),

and Df = {(η,J , ψ)} be δ-fine division of [a, b]. Then

Dp ∪ {(η,J , (1− σ)ψ)}

is a δ-fine division of [a, b].

Proof : Let x ∈ [a, b]. Since Df is a δ-fine division of [a, b], {ψ} is a partition of unity.

Thus, ∑
Df

ψ(x) = 1.

Observe that∑
Dp

φ(x) +
∑
Df

(1− σ(x)) · ψ(x) = σ(x) +
∑
Df

ψ(x)− σ(x)
∑
Df

ψ(x)

= σ(x) + 1− σ(x) · 1 = 1.

This means that {φ} ∪ {(1 − σ)ψ} is a partition of unity. Since Dp is a partial δ-fine

division of [a, b], it follows that

supp φ ⊆ I ⊆ B(ξ, δ(ξ)). (3.10)

Now, let x ∈ supp ((1 − σ) · ψ). Then (1 − σ(x)) · ψ(x) > 0 and so ψ(x) > 0. Hence,

x ∈ supp ψ. Thus, supp ((1− φ) · ψ) ⊆ supp ψ. So we have

supp ((1− σ) · ψ) ⊆ supp ψ ⊆ J ⊆ B(η, δ(η)). (3.11)

By the inclusions in 3.10 and 3.11, evidently

D ∪ {(η,J , (1− σ)ψ)}

is a δ-fine division of [a, b]. □

Lemma 3. Let f : [a, b] → X be a PU integrable function over [a, b]. If G be an open

and bounded set such that supp f ⊆ G ⊆ Rn, then the PU integral

(P)

∫
[a,b]

f · ϕ

exists for any continuously differentiable function ϕ : G → R with

0 < ϕ(x) ≤ 1, for all x ∈ G.



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 10 of 16

Proof : Fix ϵ > 0. Then we choose a gauge δ1 on [a, b] such that for any δ1-fine division

D = {(ξ, φ, I)} of [a, b], we have∥∥∥∥∑
D

f(ξ)

∫
I
φ− (P)

∫
[a,b]

f

∥∥∥∥ < ϵ.

Assume that B(x, δ1(x)) ⊆ G, for each x ∈ supp f . Put M = V (g; [a, b]) ∈ R and we

choose δ2(x) > 0 such that for any y ∈ [a, b] with y ∈ B(x, δ2(x)), we have

∥f(x)∥ · |ϕ(y)− ϕ(x)| < ϵ

3(M + 1)
. (3.12)

Take δ(x) = min{δ1(x), δ2(x)}, for all x ∈ supp f . Note that for each x ∈ supp f , if

y ∈ B(x, δ(x)), the inequality (3.12) still holds. For each x ∈ supp f , we have

∥f(x)∥ · sup{|ϕ(y)− ϕ(x)| : y ∈ B(x, δ(x))} < ϵ

3(M + 1)
(3.13)

Now, let D1 = {(ξ, φ, I)} and D2 = {(η, ψ,J)} be any δ-fine divisions of [a, b]. For each

x ∈ supp f , since 0 < ϕ(x) ≤ 1, we have∑
D1

ϕ(x) · φ(x) = ϕ(x) ·
∑
D1

φ(x) = ϕ(x) ≤ 1.

So, {ϕ · φ} is a partial partition of unity on supp f . Since ϕ(x) > 0 for all x ∈ G, it is

evident that

supp (ϕ · φ) = supp (φ).

Note that D1 is a δ-fine division of [a, b]. Then

supp (ϕ · φ) = supp φ ⊆ I ⊆ B(ξ, δ(ξ)).

Hence, {(ξ, ϕ · φ, I)} is a partial δ-fine division of [a, b].

Since 0 < ϕ(x) ≤ 1 for all supp f ,∑
D2

(1− ϕ(x)) · ψ(x) = (1− ϕ(x)) ·
∑
D2

ψ(x) = 1− ϕ(x) ≤ 1.

So, {(1− ϕ) · ψ)} is a partial partition of unity on supp f .

Moreover,

supp (1− ϕ) · ψ = supp ψ ⊆ J ⊆ B(η, δ(η)).

Hence, {(η, (1 − ϕ) · ψ,J)} is a partial δ-fine division of [a, b] which, by Lemma 2, will

further imply that {(ξ, ϕ · φ, I)} ∪ {(η, (1− ϕ) · ψ,J)} is δ-fine division of [a, b]. But f is

PU integrability over [a, b]; so,∥∥∥∥(∑
D1

f(ξ)

∫
I
ϕ · φ +

∑
D2

f(η)

∫
J
(1− ϕ) · ψ

)
− (P)

∫
[a,b]

f

∥∥∥∥ < ϵ

6
. (3.14)



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 11 of 16

Since f is PU integrable over [a, b] and D2 is a δ-fine division of [a, b], we then have∥∥∥∥∑
D2

f(η)

∫
J
ψ − (P)

∫
[a,b]

f

∥∥∥∥ < ϵ

6
. (3.15)

By (3.14) and (3.15), we have∥∥∥∥∑
D1

f(ξ)

∫
I
ϕ · φ −

∑
D2

f(η)

∫
J
ϕ · ψ

∥∥∥∥
=

∥∥∥∥(∑
D1

f(ξ)

∫
I
ϕ · φ +

∑
D2

f(η)

∫
J
(1− ϕ) · ψ

)
− (P)

∫
[a,b]

f

+ (P)

∫
[a,b]

f −
∑
D2

f(η)

∫
J
ψ

∥∥∥∥
≤

∥∥∥∥(∑
D1

f(ξ)

∫
I
ϕ · φ+

∑
D2

f(η)

∫
J
(1− ϕ) · ψ

)
− (P)

∫
[a,b]

f

∥∥∥∥
+

∥∥∥∥(P)

∫
[a,b]

f −
∑
D2

f(ξ)

∫
I
ψ

∥∥∥∥ < ϵ

6
+
ϵ

6
=
ϵ

3
,

that is, ∥∥∥∥∑
D1

f(ξ)

∫
I
ϕ · φ−

∑
D2

f(η)

∫
J
ϕ · ψ

∥∥∥∥ < ϵ

3
(3.16)

Further, by (3.13)∥∥∥∥∑
D1

f(ξ)ϕ(ξ)

∫
I
φ−

∑
D1

f(ξ)

∫
I
ϕ · φ

∥∥∥∥
=

∥∥∥∥∑
D1

f(ξ)

∫
I
ϕ(ξ) · φ−

∑
D1

f(ξ)

∫
I
ϕ · φ

∥∥∥∥
=

∥∥∥∥∑
D1

f(ξ)

[ ∫
I
ϕ(ξ) · φ−

∫
I
ϕ · φ

]∥∥∥∥
=

∥∥∥∥∑
D1

f(ξ)

[ ∫
I

(
ϕ(ξ) · φ− ϕ · φ

)]∥∥∥∥
=

∥∥∥∥∑
D1

f(ξ)

[ ∫
I

(
ϕ(ξ)− ϕ

)
· φ

]∥∥∥∥
≤

∥∥∥∥∑
D1

f(ξ) ·
∫
I
sup

{
|ϕ(ξ)− ϕ(x)| : x ∈ B(ξ, δ(ξ))

}
· φ

∥∥∥∥
≤

∥∥∥∥∑
D1

f(ξ) · sup
{
|ϕ(ξ)− ϕ(x)| : x ∈ B(ξ, δ(ξ))

}
·
∫
I
φ

∥∥∥∥



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 12 of 16

≤
∑
D1

∥∥f(ξ)∥∥ · sup
{
|ϕ(ξ)− ϕ(x)| : x ∈ B(ξ, δ(ξ))

}
·
∫
I
φ

≤ ϵ

3(M + 1)
·
∑
D1

∫
I
φ ≤ ϵ

3(M + 1)
· (M + 1) =

ϵ

3
,

that is, ∥∥∥∥∑
D1

f(ξ)ϕ(ξ)

∫
I
φ−

∑
D1

f(ξ)

∫
I
ϕ · φ

∥∥∥∥ < ϵ

3
(3.17)

In similar fashion,∥∥∥∥∑
D2

f(η)ϕ(η)

∫
J
ψ −

∑
D2

f(η)

∫
J
ϕ · ψ

∥∥∥∥
=

∥∥∥∥∑
D2

f(η)

∫
J
ϕ(η) · ψ −

∑
D2

f(η)

∫
J
ϕ · ψ

∥∥∥∥
=

∥∥∥∥∑
D2

f(η)

[ ∫
J
ϕ(η) · ψ −

∫
J
ϕ · ψ

]∥∥∥∥
=

∥∥∥∥∑
D2

f(η)

[ ∫
J

(
ϕ(η) · ψ − ϕ · ψ

)]∥∥∥∥
=

∥∥∥∥∑
D2

f(η)

[ ∫
J

(
ϕ(η)− ϕ

)
· ψ

]∥∥∥∥
≤

∥∥∥∥∑
D2

f(η) ·
∫
J
sup

{
|ϕ(η)− ϕ(x)| : x ∈ B(η, δ(η))

}
· ψ

∥∥∥∥
≤

∥∥∥∥∑
D2

f(η) · sup
{
|ϕ(η)− ϕ(x)| : x ∈ B(η, δ(η))

}
·
∫
J
ψ

∥∥∥∥
≤

∑
D2

∥∥f(η)∥∥ · sup
{
|ϕ(η)− ϕ(x)| : x ∈ B(η, δ(η))

}
·
∫
J
ψ

≤ ϵ

3
(
M + 1

) ·
∑
D2

∫
J
ψ ≤ ϵ

3
(
M + 1

) · (M + 1) =
ϵ

3
,

in other words, ∥∥∥∥∑
D2

f(η)ϕ(η)

∫
J
ψ −

∑
D2

f(η)

∫
J
ϕ · ψ

∥∥∥∥ < ϵ

3
(3.18)

Therefore, by (3.16), (3.17), and (3.18)∥∥∥∥∑
D1

f(ξ)ϕ(ξ)

∫
I
φ−

∑
D2

f(η)ϕ(η)

∫
I
ψ

∥∥∥∥



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 13 of 16

=

∥∥∥∥∑
D1

f(ξ)ϕ(ξ)

∫
I
φ−

∑
D1

f(ξ)

∫
I
ϕ · φ+

∑
D1

f(ξ)

∫
I
ϕ · φ

−
∑
D2

f(η)

∫
J
ϕ · ψ +

∑
D2

f(η)

∫
J
ϕ · ψ −

∑
D2

f(η)ϕ(η)

∫
J
ψ

∥∥∥∥
≤

∥∥∥∥∑
D1

f(ξ)ϕ(ξ)

∫
I
φ−

∑
D1

f(ξ)

∫
I
ϕ · φ

∥∥∥∥
+

∥∥∥∥∑
D1

f(ξ)

∫
I
ϕ · φ−

∑
D2

f(η)

∫
J
ϕ · ψ

∥∥∥∥
+

∥∥∥∥∑
D2

f(η)

∫
J
ϕ · ψ −

∑
D2

f(η)ϕ(η)

∫
J
ψ

∥∥∥∥
<
ϵ

3
+
ϵ

3
+
ϵ

3
= ϵ.

And the result follows. □
We now show a version of the Saks-Henstock Lemma for the PU integral.

Theorem 2. (Saks-Henstock Lemma) If f : [a, b] → X is PU integrable over [a, b],

then for every ϵ > 0, there exists a gauge δ on [a, b] such that for any δ-fine partial division

P = {(ξ, φ, I)} of [a, b], we have∥∥∥∥∥∑
P

(
f(ξ)

∫
I
φ− (P)

∫
I
fφ

)∥∥∥∥∥ < ϵ.

Proof : Fix ϵ > 0. Then choose a gauge δ on [a, b] such that whenever D = {(ξ, σ, I)} is

a δ-fine division of [a, b], we have∥∥∥∥∑
D

f(ξ)

∫
I
σ − (P)

∫
[a,b]

f

∥∥∥∥ < ϵ.

Let P = {(ξ, φ, I)} be a δ-fine partial division of [a, b] and put ϕ =
∑
P

φ. We choose a

δ1(x) ≤ δ(x) such that for any y ∈ [a, b] ∩B(x, δ1(x)), we have

∥f(x)∥|ϕ(y)− ϕ(x)| < ϵ

3
.

So, for each x ∈ supp f

∥f(x)∥ · sup
{
|ϕ(x)− ϕ(y)| : y ∈ B(x, δ1(x))

}
<
ϵ

3
. (3.19)

Now, by Lemma 3, f · (1− ϕ) is PU integrable over [a, b]. Thus, there is a gauge δ2 ≤ δ1
on [a, b] such that for every δ2-fine division D = {(ξ′, ψ,J)} of [a, b], we have∥∥∥∥∑

D

f(ξ′) · (1− ϕ(ξ′))

∫
J
ψ − (P)

∫
[a,b]

f · (1− ϕ)

∥∥∥∥



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 14 of 16

=

∥∥∥∥∑
D

f(ξ′) · (1− ϕ(ξ′))

∫
J
ψ − (P)

∫
[a,b]

f + (P)

∫
[a,b]

f ·
(∑

P

φ

)∥∥∥∥
=

∥∥∥∥∑
D

f(ξ′) · (1− ϕ(ξ′))

∫
J
ψ − (P)

∫
[a,b]

f +
∑
P

∫
I
f · φ

∥∥∥∥ < ϵ

3
,

that is, ∥∥∥∥∑
D

f(ξ′) · (1− ϕ(ξ′))

∫
J
ψ +

∑
P

∫
I
f · φ− (P)

∫
[a,b]

f

∥∥∥∥ < ϵ

3
. (3.20)

Now, since P ∪ {(ξ,J , (1 − ϕ)ψ) : (ξ,J , ψ) ∈ D} is also a δ-fine division of [a, b] and by

integrability of f , we have∥∥∥∥∑
P

f(ξ)

∫
I
φ+

∑
D

f(ξ′)

∫
J
(1− ϕ)ψ − (P)

∫
[a,b]

f

∥∥∥∥ < ϵ

3
. (3.21)

Observe that by (3.19),∥∥∥∥∑
D

f(ξ′)(1− ϕ(ξ′))

∫
J
ψ −

∑
D

f(ξ′)

∫
J
(1− ϕ)ψ

∥∥∥∥
=

∥∥∥∥∑
D

{
f(ξ′) ·

∫
J

[
ϕ− ϕ(ξ′)

]
ψ

}∥∥∥∥
≤

∥∥∥∥∑
D

{
f(ξ′) ·

∫
J
sup

{
|ϕ(x)− ϕ(ξ′)| : x ∈ B(ξ′, δ1(ξ

′))
}
· ψ

}∥∥∥∥
=

∥∥∥∥∑
D

{
f(ξ′) · sup

{
|ϕ(x)− ϕ(ξ′)| : x ∈ B(ξ′, δ1(ξ

′))
}
·
∫
J
ψ

}∥∥∥∥
≤

∑
D

{∥∥f(ξ′)∥∥ · sup
{
|ϕ(x)− ϕ(ξ′)| : x ∈ B(ξ′, δ1(ξ

′))
}
·
∫
J
ψ

}
<
ϵ

3
;

that is, ∥∥∥∥∑
D

f(ξ′)(1− ϕ(ξ′))

∫
J
ψ −

∑
D

f(ξ′)

∫
J
(1− ϕ)ψ

∥∥∥∥ < ϵ

3
. (3.22)

Therefore, by (3.20), (3.21), and (3.22)∥∥∥∥∑
P

(
f(ξ)

∫
I
φ−

∫
I
fφ

)∥∥∥∥
=

∥∥∥∥∑
P

f(ξ)

∫
I
φ − (P)

∫
[a,b]

f + (P)

∫
[a,b]

f −
∑
D

f(ξ′)(1− ϕ(ξ′))

∫
J
ψ

+
∑
D

f(ξ′)(1− ϕ(ξ′))

∫
J
ψ −

∑
D

f(ξ′)

∫
J
(1− ϕ)ψ



G. Flores, A. Flores / Eur. J. Pure Appl. Math, 18 (2) (2025), 5611 15 of 16

+
∑
D

f(ξ′)

∫
J
(1− ϕ)ψ −

∑
P

∫
I
fφ

∥∥∥∥
≤

∥∥∥∥∑
P

f(ξ)

∫
I
φ+

∑
D

f(ξ′)

∫
J
(1− ϕ)ψ − (P)

∫
[a,b]

f

∥∥∥∥
+

∥∥∥∥(P)

∫
[a,b]

f −
∑
P

∫
I
fφ−

∑
D

f(ξ′)(1− ϕ(ξ′))

∫
I
ψ

∥∥∥∥
+

∥∥∥∥∑
D

f(ξ′)(1− ϕ(ξ′))

∫
J
ψ −

∑
D

f(ξ′)

∫
J
(1− ϕ)ψ

∥∥∥∥
<
ϵ

3
+
ϵ

3
+
ϵ

3
= ϵ.

And the result follows. □

Corollary 1. In view of the conditions of Theorem 2, if in addition that X is a finite

dimensional Banach space, then for every ϵ > 0, there exists a gauge δ on [a, b] such that

for any δ-fine partial division P = {(ξ, φ, I)} of [a, b], we have∑
P

∥∥∥∥f(ξ)∫
I
φ− (P)

∫
I
fφ

∥∥∥∥ < ϵ.

Proof : Note that each norms defined on finite dimensional normed spaces are equivalent.

The proof is complete by considering each components of X, endowed with, perhaps, the

maximum norm. □

Acknowledgements

This paper is dedicated to the memory of our mentor Dr. Julius V. Benitez. The

authors would like to thank the Central Mindanao University through its Research

Office for the support of this paper.

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