Chapter 2

Henstock-Stieltjes Integral in

Banach Spaces

2.1 Introduction and preliminaries

The Henstock-Stieltjes integral is a generalized Riemann-Stieltjes integral

which has, as the name suggests, a definition similar to the Riemann-Stieltjes

integral. The Henstock integral is a natural extension of the Riemann integral

which has been developed by R. Henstock in 1960, but some reason it has not

become well-known. Many authors, in the last two decades, have studied the

notion of integrals for Banach space-valued functions. Much of the work has

been done on various integrals of function taking values in Banach spaces.

In the literature, Cao [4], firstly studied Henstock integral for Banach space-

valued functions. In 1998, Lim, et al [8], have defined Henstock-Stieltjes

integral of real-valued functions with respect to an increasing function. These

facts motivates us to study the Henstock-Stieltjes integral for Banach space-

valued functions.

In this chapter, we will define Henstock-Stieltjes integral of Banach space-

27

valued function with respect to a function of bounded variation which is

an extension of real-valued Henstock-Stieltjes integral with respect to an

increasing function. We will investigate some properties of the Henstock-

Stieltjes integral of Banach space-valued functions. The main results of this

chapter are already appeared in [11].

Throughout this chapter, (X, ‖ · ‖X) will denote a Banach space and X∗

its dual.

Let P = {(ξi, [xi, xi+1])}pi=0 be a finite collection of non-overlapping tagged

intervals in I, let f : I → X, and φ : I → R be a function of bounded

variation. The sum S(f, dφ;P) =p∑i=0

f(ξi)(φ(xi+1)−φ(xi)) is called Riemann-

Stieltjes sum of f with respect to φ.

2.2 Henstock-Stieltjes Integral

In this section we define Henstock-Stieltjes integral for Banach space-valued

functions and study some of its basic properties.

Definition 2.2.1. Let φ : I → R be a function of bounded variation. A

function f : I → X is Henstock-Stieltjes integrable with respect to φ on I,

if there exists a vector A ∈ X with the following property: for every ε > 0,

there exists a gauge δ on I such that ‖S(f, dφ;P)−A‖X < ε whenever P is

a δ-fine tagged partition of I.

For Simplicity we write A =∫If dφ.

For X = R, we observed that if φ(x) = x, then the above integral reduces

to the Henstock-Kurzweil integral [2]. If δ is a constant, then it reduces to

the Riemann-Stieltjes integral [10]. Moreover, if φ(x) = x and δ is a constant,

then it reduces to the Riemann integral [3].

The function f is Henstock-Stieltjes integrable on a measurable set E ⊆ I

28

with respect to φ if fχE is Henstock-Stieltjes integrable with respect to φ on

I.

Theorem 2.2.2. Let φ : I → R be a function of bounded variation. Suppose

f : I → X is Henstock-Stieltjes integrable with respect to φ on I. Then∥∥∥∥∫I

f dφ

∥∥∥∥X

≤ supx∈[a,b]

‖f(x)‖X · V ar(φ, I).

Proof. Let ε > 0 be given.

By assumption, there exists a gauge δ on I such that∥∥∥∥S(f, dφ;P)−∫I

f dφ

∥∥∥∥X

< ε

whenever P = {(ξi, [xi, xi+1])}pi=0 is a δ-fine partition of I.

Then ∥∥∥∥∫I

f dφ

∥∥∥∥X

=

∥∥∥∥∫I

f dφ− S(f, dφ;P) + S(f, dφ;P)

∥∥∥∥X

≤∥∥∥∥∫

I

f dφ− S(f, dφ;P)

∥∥∥∥X

+ ‖S(f, dφ;P)‖X

< ε+

p∑i=0

‖f(ξi)‖X · |φ(xi+1)− φ(xi)|

≤ ε+ supx∈[a,b]

‖f(x)‖X · V ar(φ, I).

Since ε > 0 is arbitrary, we have∥∥∥∥∫I

f dφ

∥∥∥∥X

≤ supx∈[a,b]

‖f(x)‖X · V ar(φ, I).

The following theorem shows that the set of all Henstock-Stieltjes inte-

grable functions on I forms a vector space over R.

Theorem 2.2.3. Let φ : I → R be a function of bounded variation. Let

f, g : I → X be Henstock-Stieltjes integrable with respect to φ on I. Then

29

i) For every k ∈ R, kf is Henstock-Stieltjes integrable with respect to φ

on I and ∫I

k f dφ = k

∫I

f dφ.

ii) f + g is Henstock-Stieltjes integrable with respect to φ on I and∫I

(f + g) dφ =

∫I

f dφ+

∫I

g dφ.

Proof. i) Let f : I → X be Henstock-Stieltjes integrable with respect to φ

on I.

Case I) If k = 0, the result is obvious.

Case II) k 6= 0,

For given ε > 0, there exists a gauge δ on I such that∥∥∥∥S(f, dφ;P)−∫I

f dφ

∥∥∥∥X

<ε

|k|

whenever P is a δ-fine tagged partition of I.

Therefore

|k| ·∥∥∥∥S(f, dφ;P)−

∫I

f dφ

∥∥∥∥X

< ε.

That is, ∥∥∥∥S(kf, dφ;P)− k∫I

f dφ

∥∥∥∥X

< ε.

This implies that the function kf is Henstock-Stieltjes integrable with respect

to φ on I and ∫I

k f dφ = k

∫I

f dφ.

ii) Let f, g : I → X be Henstock-Stieltjes integrable with respect to φ on I.

Then there exist a gauge δ1 on I such that∥∥∥∥S(f, dφ;P1)−∫I

f dφ

∥∥∥∥X

<ε

2(2.1)

30

whenever P1 is a δ1-fine tagged partition of I and a gauge δ2 on I such that∥∥∥∥S(g, dφ;P2)−∫I

g dφ

∥∥∥∥X

<ε

2(2.2)

whenever P2 is a δ2-fine tagged partition of I.

Define a gauge δ by δ(x) = min{δ1(x), δ2(x)}.

Now let P be a δ-fine tagged partition of I.

Then ∥∥∥∥S(f + g, dφ;P)−(∫

I

f dφ+

∫I

g dφ

)∥∥∥∥X

=

∥∥∥∥S(f, dφ;P)−∫I

f dφ+ S(g, dφ;P)−∫I

g dφ

∥∥∥∥X

≤∥∥∥∥S(f, dφ;P)−

∫I

f dφ

∥∥∥∥X

+

∥∥∥∥S(g, dφ;P)−∫I

g dφ

∥∥∥∥X

<ε

2+ε

2= ε (by (2.1) and (2.2)) .

Therefore for given ε > 0, there is a gauge δ on I such that∥∥∥∥S(f + g, dφ;P)−(∫

I

fdφ+

∫I

gdφ

)∥∥∥∥X

< ε.

whenever P is a δ-fine tagged partition of I.

Hence f + g is Henstock-Stieltjes integrable with respect to φ on I.

Theorem 2.2.4. Let φ : I → R be a function of bounded variation. Let

f : I → X and c ∈ I. If f is Henstock-Stieltjes integrable with respect to φ

on each of [a, c] and [c, b], then f is Henstock-Stieltjes integrable with respect

to φ on I and ∫I

f dφ =

∫ c

a

f dφ+

∫ b

c

f dφ.

Proof. Let f : I → X and c ∈ (a, b).

Suppose that f is Henstock-Stieltjes integrable with respect to φ on each of

31

the subintervals [a, c] and [c, b].

Let ε > 0 be given. Then there exist a gauge δ1 on [a, c] such that∥∥∥∥S(f, dφ;P1)−∫ c

a

f dφ

∥∥∥∥X

<ε

2

whenever P1 is a δ1-fine tagged partition of [a, c] and a gauge δ2 on [c, b] such

that ∥∥∥∥S(f, dφ;P2)−∫ b

c

f dφ

∥∥∥∥X

<ε

2

whenever P2 is a δ2-fine tagged partition of [c, b].

Define a gauge δ on [a, b] by

δ(x) =

min{δ1(x), c− x}, if a ≤ x < c

min{δ1(x), δ2(x)}, if x = c

min{x− c, δ2(x)}, if c < x ≤ b.

Let P be a δ-fine tagged partition of [a, b] and suppose each tag occurs only

once.

Then P must be of the form P = Pa ∪ (c, [u, v]) ∪ Pb, where the tags of Paare less than c and that of Pb are greater than c.

Let P1 = Pa ∪ (c, [u, c]) and P2 = (c, [c, v]) ∪ Pb.

Then P1 is a tagged partition of [a, c] which is δ1-fine and P2 is a tagged

partition of [c, b] which is δ2-fine.

And since

S(f, dφ;P) = S(f, dφ;P1 + P2)

= S(f, dφ;P1) + S(f, dφ;P2),

we have ∥∥∥∥S(f, dφ;P)−(∫ c

a

f dφ+

∫ b

c

f dφ

)∥∥∥∥X

32

=

∥∥∥∥S(f, dφ;P1) + S(f, dφ;P2)−∫ c

a

f dφ−∫ b

c

f dφ

∥∥∥∥X

≤∥∥∥∥S(f, dφ;P1)−

∫ c

a

f dφ

∥∥∥∥X

+

∥∥∥∥S(f, dφ;P2)−∫ b

c

f dφ

∥∥∥∥X

<ε

2+ε

2= ε.

Therefore for given ε > 0, there is a gauge δ on [a, b] such that∥∥∥∥S(f, dφ;P)−(∫ c

a

f dφ+

∫ b

c

f dφ

)∥∥∥∥X

< ε

whenever P is a δ-fine tagged partition of [a, b].

This shows that f is Henstock-Stieltjes integrable with respect to φ on [a, b]

and ∫I

f dφ =

∫ c

a

f dφ+

∫ b

c

f dφ.

Theorem 2.2.5. Let f : I → X be such that f = 0 almost everywhere on

I. Let φ : I → R be a function of bounded variation such that φ ∈ C1(I).

Then f is Henstock-Stieltjes integrable with respect to φ on I and∫I

f dφ = 0.

Proof. Let {an : n ∈ Z+} = {x ∈ I : f(x) 6= 0} and let ε > 0.

Since φ ∈ C1(I), there exists M such that |φ′(x)| ≤M for all x ∈ I.

By mean-value theorem [3], there exists θi ∈ (xi, xi+1) such that φ(xi+1) −

φ(xi) = φ′(θi)(xi+1 − xi).

Define a gauge δ by

δ(x) =

1, if x ∈ I − {an : n ∈ Z+}

ε‖f(an)‖XM 2n+1 , if x = an.

Suppose that P = {(ξi, [xi, xi+1])}pi=0 is a δ-fine partition of I and assume

each tag occurs only once.

33

Let π be the set of all indices i such that ξi ∈ {an : n ∈ Z+} and for each

i ∈ π, choose ni so that ξi = ani . Then

‖S(f, dφ;P)‖X =

∥∥∥∥∥∑i∈π

f(ξi)(φ(xi+1)− φ(xi))

∥∥∥∥∥X

=

∥∥∥∥∥∑i∈π

f(ξi)φ′(θi)(xi+1 − xi)

∥∥∥∥∥X

≤∑i∈π

‖f(ξi)‖X · |φ′(θi)| · |xi+1 − xi|

≤∑i∈π

‖f(ani)‖X ·M · 2 δ(ξi)

=∑i∈π

‖f(ani)‖X ·M · 2ε

‖f(ani)‖XM 2ni+1

=∑i∈π

ε

2ni= ε.

Therefore ‖S(f, dφ;P)‖X < ε.

This shows that the function f : I → X is Henstock-Stieltjes integrable with

respect to φ on I and ∫I

f dφ = 0.

Just as in case of the Henstock integral, the Cauchy criterion holds for

the Henstock-Stieltjes integral of Banach space-valued functions.

Theorem 2.2.6. Let φ : I → R be a function of bounded variation. Then a

function f : I → X is Henstock-Stieltjes integrable with respect to φ on I if

and only if for every positive ε, there exists a gauge δ on I such that

‖S(f, dφ;P1)− S(f, dφ;P2)‖X < ε

whenever P1 and P2 are tagged partitions of I that are δ-fine.

34

Proof. Let ε > 0 be given.

Suppose f is Henstock-Stieltjes integrable with respect to φ on I.

Then there exists a gauge δ on I such that∥∥∥∥S(f, dφ;P1)−∫I

f dφ

∥∥∥∥X

<ε

2,

∥∥∥∥S(f, dφ;P2)−∫I

f dφ

∥∥∥∥X

<ε

2

whenever P1 and P2 are two δ-fine partitions of I.

Then ‖S(f, dφ;P1)− S(f, dφ;P2)‖X

=

∥∥∥∥S(f, dφ;P1)−∫I

f dφ+

∫I

f dφ− S(f, dφ;P2)

∥∥∥∥X

≤∥∥∥∥S(f, dφ;P1)−

∫I

f dφ

∥∥∥∥X

+

∥∥∥∥S(f, dφ;P2)−∫I

f dφ

∥∥∥∥X

<ε

2+ε

2= ε.

Therefore

‖S(f, dφ;P1)− S(f, dφ;P2)‖X < ε.

Hence the Cauchy criterion is satisfied.

Conversely, suppose that the Cauchy criterion is satisfied.

For each positive integer n, choose a gauge δn on I such that

‖S(f, dφ;P1)− S(f, dφ;P2)‖X <1

n

whenever P1 and P2 are δn-fine partitions of I.

Let us assume that the sequence {δn} is nonincreasing.

For each n, let Pn be a δn-fine partition of I.

We have for m,n ≥ N , N ∈ N,

‖S(f, dφ; Pn)− S(f, dφ;Pm)‖X <1

N. (2.3)

Therefore the sequence {S(f, dφ;Pn)} is a Cauchy sequence in X and hence

convergent.

35

Let L be the limit of that sequence and ε > 0 be given.

Choose a positive integer N such that 1N< ε

2and

‖S(f, dφ;Pn)− L‖X <ε

2, ∀ n ≥ N. (2.4)

Let P be a δN -fine partition of I.

Then ‖S(f, dφ;P)− L‖X

= ‖S(f, dφ;P)− S(f, dφ;PN) + S(f, dφ;PN)− L‖X

≤ ‖S(f, dφ;P)− S(f, dφ;PN)‖X + ‖S(f, dφ;PN)− L‖X

<1

N+ε

2(by (2.3) and (2.4))

<ε

2+ε

2= ε.

Therefore ‖S(f, dφ;P)−L‖X < ε. This shows that the function f is Henstock-

Stieltjes integrable with respect to φ on I.

As a converse of the Theorem (2.2.4), we shall state the following result.

Theorem 2.2.7. Let φ : I → R be a function of bounded variation. If

f : I → X is Henstock-Stieltjes integrable with respect to φ on I, then f is

also Henstock-Stieltjes integrable with respect to φ on every subinterval of I.

Now the following theorem shows that the Henstock-Stieltjes integral re-

ally exists for certain functions f .

Theorem 2.2.8. Let φ : I → R be a function of bounded variation. Suppose

f : I → X is a regulated function. Then the Henstock-Stieltjes integral∫If dφ exists.

Proof. Since f : I → X is regulated, for every ε > 0, there exists a partition

P : a = x0 < x1 < . . . < xp+1 = b of I such that ‖f(ξi) − f(ηi)‖X < ε for

ξi, ηi ∈ (xi, xi+1), i = 0, 1, 2, . . . , p [5].

36

Let δ be a gauge defined on I and P = {(ξi, [xi, xi+1])}pi=0 be a δ-fine partition

of I. Then we have∥∥∥∥S(f, dφ; P)−∫I

f dφ

∥∥∥∥X

=

∥∥∥∥∥p∑i=0

f(ξi)(φ(xi+1)− φ(xi))−p∑i=0

∫ xi+1

xi

f dφ

∥∥∥∥∥X

≤ ‖f(ξi)− f(ηi)‖X · supxi, xi+1∈I

p∑i=0

|φ(xi+1)− φ(xi)|

< ε · V ar(φ, I) = ε1.

Hence for every ε1 > 0, there is a gauge δ on I such that∥∥∥∥S(f, dφ; P)−∫I

f dφ

∥∥∥∥X

< ε1

whenever P = {(ξi, [xi, xi+1])}pi=0 is a δ-fine partition of I.

Which means that the integral∫If dφ exists.

2.3 Convergence Theorems

In this section first we shall present uniform convergence theorem for the

Henstock-Stieltjes integral of Banach space-valued functions. While uniform

convergence is a very severe restriction, it remains an important mode of

convergence in the theory of integration.

Theorem 2.3.1. Let φ : I → R be a function of bounded variation. Let {fn}

be a sequence of X-valued functions that are Henstock-Stieltjes integrable

with respect to φ on I. Assume that the sequence {fn} converges uniformly

to f on I, i.e., limn→∞

‖fn(x) − f(x)‖X −→ 0 uniformly on I. Then f is

Henstock-Stieltjes integrable with respect to φ on I and∫I

f(x) dφ(x) = limn→∞

∫I

fn(x) dφ(x).

37

Proof. Let ε > 0 be given.

Since the sequence {fn} converges to f uniformly on I, there exists a positive

integer N such that for any n > N and x ∈ I,

‖fn(x)− f(x)‖X <ε

6 [V ar(φ, I) + 1].

So for any n,m > N and x ∈ I, we have

‖fn(x)− fm(x)‖X = ‖fn(x)− f(x) + f(x)− fm(x)‖X

≤ ‖fn(x)− f(x)‖X + ‖f(x)− fm(x)‖X

<ε

6 [V ar(φ, I) + 1]+

ε

6 [V ar(φ, I) + 1]

<ε

3 [V ar(φ, I) + 1].

Now∥∥∥∥∫I

fn(x) dφ(x)−∫I

fm(x) dφ(x)

∥∥∥∥X

=

∥∥∥∥∫I

[fn(x)− fm(x)] dφ(x)

∥∥∥∥X

≤ supx∈I‖fn(x)− fm(x)‖X · V ar(φ, I)

<ε

3 [V ar(φ, I) + 1]· V ar(φ, I)

<ε

3.

That is, ∥∥∥∥∫I

fn(x) dφ(x)−∫I

fm(x) dφ(x)

∥∥∥∥X

<ε

3. (2.5)

Therefore{∫

Ifn(x) dφ(x)

}forms a Cauchy sequence in X and hence conver-

gent.

Let A ∈ X be such that limn→∞

∫Ifn(x) dφ(x) = A.

Let N1 ∈ N be such that for n > N1, we have∥∥∥∥∫I

fn(x) dφ(x)− A∥∥∥∥X

<ε

3.

38

Let m > max{N,N1}.

Since the integral∫Ifm(x) dφ(x) exists, there is a gauge δ on I such that∥∥∥∥S(fm, dφ;P)−

∫I

fm(x) dφ(x)

∥∥∥∥X

<ε

3(2.6)

whenever P = {(ξi, [xi, xi+1])}pi=0 is a δ-fine partition of I.

Then for such a δ-fine partition P ,

‖S(f, dφ;P)− A‖X

=

∥∥∥∥∥p∑i=0

f(ξi)(φ(xi+1)− φ(xi))− A

∥∥∥∥∥X

=

∥∥∥∥∥p∑i=0

f(ξi)(φ(xi+1)− φ(xi))−p∑i=0

fm(ξi)(φ(xi+1)− φ(xi))

+

p∑i=0

fm(ξi)(φ(xi+1)− φ(xi))−∫I

fm(x) dφ(x)

+

∫I

fm(x) dφ(x)− A∥∥∥∥X

≤

∥∥∥∥∥p∑i=0

f(ξi)(φ(xi+1)− φ(xi))−p∑i=0

fm(ξi)(φ(xi+1)− φ(xi))

∥∥∥∥∥X

+

∥∥∥∥∥p∑i=0

fm(ξi)(φ(xi+1)− φ(xi))−∫I

fm(x) dφ(x)

∥∥∥∥∥X

+

∥∥∥∥∫I

fm(x) dφ(x)− A∥∥∥∥X

=

∥∥∥∥∥p∑i=0

[f(ξi)− fm(ξi)](φ(xi+1)− φ(xi))

∥∥∥∥∥X

+ε

3+ε

3

(by (2.5) and (2.6)).

Now consider ∥∥∥∥∥p∑i=0

[f(ξi)− fm(ξi)](φ(xi+1)− φ(xi))

∥∥∥∥∥X

≤p∑i=0

‖f(ξi)− fm(ξi)‖X · |φ(xi+1)− φ(xi)|

39

≤ maxi‖f(ξi)− fm(ξi)‖X ·

p∑i=0

|φ(xi+1)− φ(xi)|

= maxi‖f(ξi)− fm(ξi)‖X · V ar(φ, I)

<ε

3.

Hence ‖S(f, dφ;P)−A‖X < ε which means that the Henstock-Stieltjes inte-

gral∫If(x) dφ(x) exists and∫

I

f(x) dφ(x) = limn→∞

∫I

fn(x) dφ(x).

We observe that defining an integral is in fact a certain “limiting process”.

Therefore the problem of the convergence theorem becomes the problem of

interchanging of two “limiting process”. It is commonly accepted and known

that two limits can be interchanged if one of the limit is uniform [1]. The

first possibility when the limit in “ limn→∞

fn(x) = f(x) for x ∈ I”is uniform

leads to a convergence result (Theorem (2.3.1)), which is well known even

for Riemann-Stieltjes integral [10], is far-reaching sufficient condition. The

other possibility is given by the following theorem in which the notion of

equi-integrability of the sequence {fn} expresses the uniformity of the limit

with respect to n of the “limiting process”of integration.

The notion of equi-integrability for the Henstock-Kurzweil integral of real-

valued functions was first introduced by J. Kurzweil [6]. Following his argu-

ment, we define the notion of equi-integrability for Henstock-Stieltjes integral

of Banach space-valued functions as follows.

Definition 2.3.2. A collection F of functions f : I → X is called Henstock-

Stieltjes equi-integrable with respect to φ on I if there exists a gauge δ on I

40

such that for every ε > 0, there exists a δ-fine partition P of I such that∥∥∥∥S(f, dφ;P)−∫I

f dφ

∥∥∥∥X

< ε for each f ∈ F.

Using the notion of Henstock-Stieltjes equi-integrability, we have the fol-

lowing convergence theorem for the Henstock-Stieltjes integral.

Theorem 2.3.3. Let φ : I → R be a function of bounded variation. Let

{fn} be a sequence of X-valued functions that are Henstock-Stieltjes equi-

integrable with respect to φ on I. If {fn} converges pointwise to f on I, then

f is Henstock-Stieltjes integrable with respect to φ on I and

limn→∞

∫I

fn dφ =

∫I

f dφ.

Proof. For ε > 0, by definition of Henstock-Stieltjes equi-integrability of {fn}

with respect to φ, we have∥∥∥∥S(fn, dφ;P)−∫I

fn dφ

∥∥∥∥X

< ε for all n ∈ N

whenever P is a δ-fine partition of I.

The integrability of f is a consequence of a simple limit. If the partition

P is fixed, then the pointwise convergence fn → f yields the sequence

{S(fn, dφ;P)} converges towards S(f, dφ;P).

Also by definition, we deduce∥∥∥∥S(f, dφ;P)−∫I

f dφ

∥∥∥∥X

< ε.

Then∥∥∥∥∫I

fn dφ−∫I

f dφ

∥∥∥∥X

≤∥∥∥∥∫

I

fn dφ− S(fn, dφ;P)

∥∥∥∥X

+ ‖S(fn, dφ;P)

− S(f, dφ;P)‖X +

∥∥∥∥S(f, dφ;P)−∫I

f dφ

∥∥∥∥X

< 3 ε for large n.

41

Therefore ∥∥∥∥∫I

fn dφ−∫I

f dφ

∥∥∥∥X

< 3 ε for large n.

Hence

limn→∞

∫I

fn dφ =

∫I

f dφ.

Remark 2.3.4. Note that a pointwise convergent sequence of HS-equi-integrable

Banach space-valued functions does not converges uniformly to the pointwise

limit.

For example:

1) Let fn : [0, 1]→ X, n ∈ N be defined as

fn(x) =

a, if x = 1

nfor some n ∈ N

a, if x = 0

0, otherwise,

where a is any vector in X.

By definition fn(0) = a. So limn→∞

fn(0) = a. Therefore for given ε > 0, there

exists N1 = 1 ∈ N such that ‖fn(0)− a‖X < ε, for all n ≥ 1.

And, fn(1) = 0 if n 6= 1. So limn→∞

fn(1) = 0. Therefore for given ε > 0, there

exists N2 = 2 ∈ N such that ‖fn(1)‖X < ε, for all n ≥ 2.

Hence the sequence {fn} does not converge uniformly. However this sequence

converges pointwise to a HK-integrable function.

Also, this sequence is HS-equi-integrable on [0, 1].

2) Let gn : [0, 1]→ l2(N) be defined as

gn(x) =

x−1 exp(−nx)un, if x ∈ (0, 1]

0, if x = 0,

42

where un is the nth unit vector in l2(N).

Take φ(x) = cosx. Then φ is of bounded variation function and dφ =

− sinx dx.

The sequence {gn} converges pointwise to 0 but not uniformly.

If possible suppose it converges uniformly. Then given 12< ε < 1, there exists

N ∈ N such that ‖x−1 exp(−nx)un‖l2(N) < ε for n ≥ N and for all x ∈ [0, 1].

Now consider x ∈ (0, 1) such that x < 1N

ln(1ε). By choice of ε, such x exists.

Then Nx < ln(1ε). So ε < exp(−Nx).

Hence ‖x−1 exp(−Nx)uN‖l2(N) ≥ ‖exp(−Nx)uN‖l2(N) > ε, which is contra-

diction.

Now ∫ 1

0

fn dφ =

∫ 1

0

x−1 exp(−nx)un(− sinx)dx

= −∫ 1

0

exp(−nx)un sinx

xdx.

Then the sequence{

exp(−nx)un sinxx

}is equi-integrable on [0, 1] [7].

Theorem 2.3.5. Suppose f : I → X is a regulated function and {φn} is a

sequence of bounded variation functions on I which is two norm convergent

to φ in BV . Then∫If dφ exists and

limn→∞

∫I

f dφn =

∫I

f dφ.

Proof. Since {φn} is two norm convergent to φ in BV and φ ∈ BV , so∫If dφ

exists.

Also f is integrable with respect to the sequence {φn} on I, then for every

ε > 0, there exists a gauge δ on I such that for every δ-fine partition P of I,

we have ∥∥∥∥S(f, dφ;P)−∫I

f dφ

∥∥∥∥X

< ε and

43

∥∥∥∥S(f, dφn;P)−∫I

f dφn

∥∥∥∥X

< ε for every n ∈ N.

Since {φn} converges uniformly to φ on I, there exists a δ-fine partition P0

of I such that

‖S(f, dφn;P0)− S(f, dφ;P0)‖X < ε for large n.

Therefore, we have∥∥∥∥∫I

f dφn −∫I

f dφ

∥∥∥∥X

=

∥∥∥∥∫I

f dφn − S(f, dφn;P0) + S(f, dφn;P0)

−S(f, dφ;P0) + S(f, dφ;P0)−∫I

f dφ

∥∥∥∥X

≤∥∥∥∥∫

I

f dφn − S(f, dφn;P0)

∥∥∥∥X

+ ‖S(f, dφn;P0)

−S(f, dφ;P0)‖X +

∥∥∥∥S(f, dφ;P0)−∫I

f dφ

∥∥∥∥X

< 3 ε for large n.

Hence

limn→∞

∫I

f dφn =

∫I

f dφ.

Theorem 2.3.6. Suppose f : I → X is a regulated function. If {φk} is a

sequence of bounded variation functions such that V ar(φk, I) ≤M for every

k and for some constant M , then f is Henstock-Stieltjes equi-integrable with

respect to the sequence {φk}.

Proof. Since f : I → X is regulated, for every ε > 0, there exists a partition

P0 : a = x0 < x1 < . . . < xp+1 = b of I such that ‖f(ξi) − f(ηi)‖X < ε for

ξi, ηi ∈ (xi, xi+1), i = 0, 1, . . . , p [5].

44

Fix i and set

φ∗k(x) =

φk(x

+i ), x = xi

φk(x), x ∈ (xi, xi+1)

φk(x−i+1), x = xi+1.

Then we have∫ xi+1

xi

f dφk =

∫ xi+1

xi

f dφ∗k + f(xi)(φk(x+i )− φk(xi))

+ f(xi+1)(φk(xi+1)− φk(x−i+1)).

Define a gauge δ on I such that every δ-fine partition P = {(ξi, [xi, xi+1]}pi=0

of I is finer than P0.

Let P be any δ-fine partition of I. Then we can write P = P1∪P2∪ . . .∪Pn,

where Pi is a δ-fine partition of [xi, xi+1].

So we have∥∥∥∥S(f, dφk;Pi)−∫ xi+1

xi

f dφk

∥∥∥∥X

=

∥∥∥∥S(f, dφ∗k;Pi)−∫ xi+1

xi

f dφ∗k

∥∥∥∥X

≤ ε · V ar(φk, [xi, xi+1]).

Hence for every δ-fine partition P = {(ξi, [xi, xi+1])}pi=0 of I,∥∥∥∥S(f, dφk;P)−∫I

f dφk

∥∥∥∥X

=

∥∥∥∥∥p∑i=0

f(ξi)(φk(xi+1)− φk(xi))

−p∑i=0

∫ xi+1

xi

f dφk

∥∥∥∥∥X

≤p∑i=0

∥∥∥∥S(f, dφk;Pi)−∫ xi+1

xi

f dφk

∥∥∥∥X

≤ ε · V ar(φk, I) ≤ ε ·M = ε1.

Thus for every ε1 > 0, there is a gauge δ on I such that for every δ-fine

partition P = {(ξi, [xi, xi+1])}pi=0 of I, we have∥∥∥∥S(f, dφk;P)−∫I

f dφk

∥∥∥∥X

< ε1 for every k.

45

Hence f is equi-integrable with respect to the sequence {φk} on I.

Theorem 2.3.7. Let φ : I → R be a function of bounded variation. Let

f : I → X be Henstock-Stieltjes integrable with respect to φ on I. Then for

any x∗ ∈ X∗, the function x∗f is Henstock-Stieltjes integrable with respect

to φ on I and ∫I

x∗ f dφ = x∗∫I

f dφ.

Also {x∗f : x∗ ∈ B(X∗)} is Henstock-Stieltjes equi-integrable with respect

to φ on I, where B(X∗) is unit ball in X∗.

Proof. Let ε > 0 be given.

Since f : I → X is Henstock-Stieltjes integrable with respect to φ on I, there

exists a gauge δ on I such that∥∥∥∥S(f, dφ;P)−∫I

f dφ

∥∥∥∥X

< ε

whenever P is a δ-fine partition of I.

Hence for any x∗ ∈ X∗, we have∣∣∣∣S(x∗f, dφ;P)− x∗∫I

f dφ

∣∣∣∣ = ‖x∗‖X∗ ·∥∥∥∥S(f, dφ;P)−

∫I

f dφ

∥∥∥∥X

< ‖x∗‖X∗ · ε.

Therefore there exists a gauge δ on I such that∣∣∣∣S(x∗f, dφ;P)− x∗∫I

f dφ

∣∣∣∣ < ε1

whenever P is δ-fine partition of I.

Hence the function x∗f is Henstock-Stieltjes integrable with respect to φ on

I.

If x∗ ∈ B(X∗), then as above∣∣∣∣S(x∗f, dφ;P)− x∗∫I

f dφ

∣∣∣∣ < ε for any x∗ ∈ B(X∗).

46

So the set {x∗f : x∗ ∈ B(X∗)} is Henstock-Stieltjes equi-integrable with

respect to φ on I.

2.4 Continuity

In this section first we shall establish a result, known as Saks-Henstock

lemma, which is useful for any advanced theory of integration based on

Riemann-type integral sums.

Theorem 2.4.1. (Saks-Henstock lemma) Let φ : I → R be a function

of bounded variation. Let f : I → X be Henstock-Stieltjes integrable with

respect to φ on I and ε > 0. Then

p∑i=0

∥∥∥∥(φ(xi+1)− φ(xi)) f(ξi)−∫ xi+1

xi

f dφ

∥∥∥∥X

< ε

whenever P = {(ξi, [xi, xi+1])}pi=0 is a δ-fine partition of I.

Proof. Suppose f : I → X is Henstock-Stieltjes integrable with respect to φ

on I.

Then by definition, there exists a vector A =∫If dφ ∈ X with the following

property: for every ε > 0, there exists a gauge δ on I such that∥∥∥∥S(f, dφ;P)−∫I

f dφ

∥∥∥∥X

< ε

whenever P is a δ-fine partition of I.

Let {δε}ε>0 be a family of gauges associated with the definition of integrability

of f with respect to φ.

We assume that δε ≤ δ for every ε > 0.

For every integer i ∈ (0, 1, . . . , p) and ε > 0, let {(ξij, [xij, xij+1])} be a tagged

partition of [xi, xi+1] which is δε-fine.

47

Now we can merge those p partitions to provide a tagged partition {ξi, I i}

of I which is δε-fine.

Also we can build a tagged partition {ζ i, I i} by merging the tagged partitions

{(ζ ij, [xij, xij+1])}, where for every i, the sequence of tags {ζ i} is a repetition

of the tag ξi, obviously {ζ i, I i} is δε-fine.

Now for every ε, the tagged partitions P i1 = {ξi, I i} and P i2 = {ζ i, I i} are

δε-fine.

Since the function f is Henstock-Stieltjes integrable with respect to φ on

I, it is integrable on every subinterval of I. Hence the Cauchy criterion is

satisfied, That is,

∥∥S(f, dφ;P i1)− S(f, dφ;P i2)∥∥X<ε

pfor each i, 0 ≤ i ≤ p.

Thereforep∑i=0

∥∥S(f, dφ;P i1)− S(f, dφ;P i2)∥∥X< ε.

That is,

p∑i=0

∥∥∥∥∥∑j

(φ(xij+1)− φ(xij)

)f(ξij)−

∑j

(φ(xij+1)− φ(xij)

)f(ζ ij)

∥∥∥∥∥X

< ε.

For each i, 0 ≤ i ≤ p,∑j

∥∥(φ(xij+1)− φ(xij))f(ξij)−

(φ(xij+1)− φ(xij)

)f(ζ ij)

∥∥X<ε

p.

By collecting the packets, we find that

p∑i=0

∑j

∥∥(φ(xij+1)− φ(xij))f(ξij)−

(φ(xij+1)− φ(xij)

)f(ζ ij)

∥∥X< ε.

But, we have

p∑i=0

∥∥∥∥∥∑j

(φ(xij+1)− φ(xij)

)f(ξij)−

∑j

(φ(xij+1)− φ(xij)

)f(ζ ij)

∥∥∥∥∥X

< ε.

48

Also as ε goes to 0, we get∑j

(φ(xij+1)− φ(xij)

)f(ζ ij) −→

∫ xi+1

xi

f dφ.

Thenp∑i=0

∥∥∥∥∥∑j

(φ(xij+1)− φ(xij)

)f(ξij)−

∫ xi+1

xi

f dφ

∥∥∥∥∥X

< ε.

But∑

j

(φ(xij+1)− φ(xij)

)f(ξij) = (φ(xi+1)− φ(xi)) f(ξi).

Therefore, we havep∑i=0

∥∥∥∥(φ(xi+1)− φ(xi)) f(ξi)−∫ xi+1

xi

f dφ

∥∥∥∥X

< ε.

Using the above theorem, we can prove the continuity of the function

F : t 7→∫ taf dφ. In fact, we have a classical stronger result that the function

F is absolutely continuous.

Theorem 2.4.2. Let φ : I → R be a function of bounded variation such

that φ ∈ C1(I). Let f : I → X be Henstock-Stieltjes integrable with respect

to φ on I. Then the function t 7→∫ taf dφ is absolutely continuous on I.

Proof. Let ε > 0 be given and δ be a gauge associated with the definition of

integrability of f with respect to φ on I.

Let P = {(ξi, [xi, xi+1])}pi=0 be a δ-fine tagged partition of I.

Let ([rk, sk])0≤k≤q be a finite collection of disjoint subintervals of I such that∑qk=0(sk − rk) < η.

Set M = 1 + max0≤k≤q

(‖f(ξk)‖X) and η = ε2MB

, |φ′(x)| ≤ B for all x ∈ I.

Repeating the tags ξk when necessary, we can include the points rk and sk

into the partition, and obtain a new partition which is δ-fine. Then we find,

by previous result and the triangle inequality, thatq∑

k=0

∥∥∥∥∫ sk

rk

f dφ

∥∥∥∥X

=

q∑k=0

‖(φ(sk)− φ(rk)) f(ξk)− (φ(sk)− φ(rk)) f(ξk)

49

+

∫ sk

rk

f dφ

∥∥∥∥X

≤q∑

k=0

∥∥∥∥(φ(sk)− φ(rk)) f(ξk)−∫ sk

rk

f dφ

∥∥∥∥X

+

q∑k=0

‖(φ(sk)− φ(rk)) f(ξk)‖X . (2.7)

Since the function f is integrable with respect to φ on I, we have

q∑k=0

∥∥∥∥(φ(sk)− φ(rk)) f(ξk)−∫ sk

rk

f dφ

∥∥∥∥X

<ε

2

and since φ is continuous on I, by mean value theorem [3], φ(sk) − φ(rk) =

φ′(θk)(sk − rk) for some θk ∈ (rk, sk).

Therefore

q∑k=0

‖(φ(sk)− φ(rk)f(ξk))‖X =

q∑k=0

‖φ′(θk)(sk − rk)f(ξk)‖X

=

q∑k=0

|φ′(θk)| · ‖f(ξk)‖X · (sk − rk)

≤q∑

k=0

B ·M · (sk − rk)

< B ·M · ε

2MB=ε

2.

Hence equation (2.7) becomes

q∑k=0

∥∥∥∥∫ sk

rk

f dφ

∥∥∥∥X

<ε

2+ε

2= ε.

That is,q∑

k=0

∥∥∥∥∫ sk

a

f dφ−∫ rk

a

f dφ

∥∥∥∥X

< ε.

Thus for every positive ε, there exists η > 0 such that

q∑k=0

∥∥∥∥∫ sk

a

f dφ−∫ rk

a

f dφ

∥∥∥∥X

< ε

50

whenever {[rk, sk]}qk=0 is a finite collection of disjoint intervals that have end

points in I and satisfy∑q

k=0(sk − rk) < η.

Hence the function F : t 7→∫ taf dφ is absolutely continuous on I.

Theorem 2.4.3. Let φ : I → R be a function of bounded variation such

that φ ∈ C1(I). Let {fn} be a sequence of X-valued Henstock-Stieltjes

integrable functions with respect to φ on I. Then the sequence {Fn(t)},

Fn(t) =∫ tafn dφ, is uniformly absolutely continuous on I = [a, b].

Proof. Let ε > 0 be given. Then by hypothesis, for each n ∈ N, there exists

a gauge δ on I such that for every δ-fine partition P = {(ξi, [xi, xi+1])}pi=0 of

I, we have ∥∥∥∥S(fn, dφ;P)−∫I

fn dφ

∥∥∥∥X

< ε.

Let {[rk, sk]}0≤k≤q be a finite collection of disjoint subintervals of I with∑qk=0(sk − rk) < η.

Set M = 1+ max0≤k≤q, n≥n0

(‖fn(ξk)‖X) for some positive integer n0 and η = εMB

,

|φ′(x)| ≤ B for all x ∈ I.

Repeating the tags ξk when necessary, we can include the points rk and sk

into the partition and obtain a new partition Q ={

(ξ′i, [x

′i, x

′i+1])

}qi=0

which

is also δ-fine.

Then by Saks-Henstock lemma, we can write

q∑k=0

∥∥∥∥∥ ∑i, rk<xi<sk

(φ(x

′

i+1)− φ(x′

i))fn(ξ

′

i)−∫ sk

rk

fn dφ

∥∥∥∥∥X

< ε for all n ∈ N.

Then

q∑k=0

‖Fn(sk)− Fn(rk)‖X =

q∑k=0

∥∥∥∥∫ sk

a

fn dφ−∫ rk

a

fn dφ

∥∥∥∥X

=

q∑k=0

∥∥∥∥∫ sk

rk

fn dφ

∥∥∥∥X

=

q∑k=0

∥∥∥∥∥ ∑i, rk<xi<sk

(φ(x

′

i+1)− φ(x′

i))fn(ξ

′

i)

51

−∑

i, rk<xi<sk

(φ(x

′

i+1)− φ(x′

i))fn(ξ

′

i) +

∫ sk

rk

fn dφ

∥∥∥∥∥X

≤q∑

k=0

∥∥∥∥∥ ∑i, rk<xi<sk

(φ(x

′

i+1)− φ(x′

i))fn(ξ

′

i)

∥∥∥∥∥X

+

q∑k=0

∥∥∥∥∥ ∑i, rk<xi<sk

(φ(x

′

i+1)− φ(x′

i))fn(ξ

′

i)−∫ sk

rk

fn dφ

∥∥∥∥∥X

≤q∑

k=0

∥∥∥∥∥ ∑i, rk<xi<sk

(φ(x

′

i+1)− φ(x′

i))fn(ξ

′

i)

∥∥∥∥∥X

+ ε

=

q∑k=0

∥∥∥∥∥ ∑i, rk<xi<sk

φ′(θi)(x

′

i+1 − x′

i) fn(ξ′

i)

∥∥∥∥∥X

+ ε

≤q∑

k=0

(sk − rk) ·MB + ε

< ε+ ε = 2 ε.

Thus for n ≥ n0, we get

q∑k=0

‖Fn(sk)− Fn(rk)‖X ≤ 2 ε whenever

q∑k=0

(sk − rk) < η.

Since Fk, k = 0, 1, 2, . . . , n0 − 1, are absolutely continuous, by the above

inequality, we can find η > 0 such that if∑q

k=0(sk − rk) < η, then

q∑k=0

‖Fn(sk)− Fn(rk)‖X ≤ ε for every n ∈ N.

Hence the sequence {Fn(t)} is uniformly absolutely continuous on I.

2.5 Equivalence with Bochner-Stieltjes inte-

gral

In this section, we shall establish the equivalence of Henstock-Stieltjes inte-

gral and Bochner-Stieltjes integral.

52

Definition 2.5.1. A measurable function f : I → X is called Bochner-

Stieltjes integrable with respect to φ if there exists a sequence {fn} of simple

functions such that limn→∞

∫I‖fn − f‖X dφ = 0.

Then we write ∫I

f dφ = limn→∞

∫I

fn dφ.

Note that the integral of f with respect to φ is independent on the sequence

{fn}.

Theorem 2.5.2. Let φ : I → R be a function of bounded variation. A

function f : I → X is Henstock-Stieltjes integrable with respect to φ if and

only if it is Bochner-Stieltjes integrable with respect to φ and the integral

coincides.

Proof. We shall use the following result for integration of Banach space-

valued functions [9]:

A function f : I → X is measurable if and only if it is the almost everywhere

uniform limit of a sequence of countably valued measurable functions.

Using this we can construct a sequence {gn} of simple functions such that∫I

‖f − gn‖X dφ <1

nand ‖gn‖X ≤ ‖f‖X +

1

nfor every n ≥ 1.

Also, since limn→∞

∫I‖f−gn‖X dφ = 0, we can choose a subsequence {gnk} such

that gnk → f .

Now gnk are integrable and ‖gnk‖X ≤ ‖f‖X + 1nk

, by dominated convergence

theorem, f is Henstock-Stieltjes integrable with respect to φ.

On the contrary, suppose f is Henstock-Stieltjes integrable with respect to

φ.

Then f is measurable and hence ‖f‖X is Lebesgue-Stieltjes integrable with

respect to φ.

Consequently, f is Bochner-Stieltjes integrable with respect to φ.

53

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55