Chapter 1

Introduction

1.1 Historical Review

The integration is basically a form of addition, it is addition of infinitely

many quantities. But it is different from a series, for a series deals with an

infinite discrete sum, while an integral deals with an infinite continuous sum.

In embryo, the theory of integration was developed out of the need to cre-

ate a general method for finding areas, volumes and centers of gravity. This

method was first employed by Greek Scholar Archimedes (287-211 BCE).

However, the systematic development of this method began only in the 17th

century in the works of Cavalieri (1598-1647), Torricelli (1608-1647), Fermat

(1601-1665), Pascal (1623-1662) and other scholars. In 1659, Isaac Barrow

(1630-1677), a pupil of Newton, established a connection between the prob-

lem of finding an area and that of finding a tangent.

The theory of integral has been an important part of mathematical analy-

sis since the 17th century when Gottfried Wilhelm Leibniz and Isaac Newton,

abstracted the above connection in the form of ‘fundamental theorem of cal-

culus’, which says that area is essentially the same as taking an antiderivative.

1

Of all the various definitions of integration that would survive a modern

critical scrutiny, by far the simplest and most intuitive is that which was given

at the beginning of the modern era by Augustin-Louis Cauchy (1789-1857).

He investigated a constructive definition of integrals of continuous functions.

In 1821, he wrote an important book [7], based on his lectures in analysis

given at the Ecole polytechnique and other Paris colleges. In which he set

forth a new concept of continuity (which we call ‘uniform continuity’ today)

which has remained standard ever since. In [8], he defined the definite integral

of a continuous function over a closed interval. Using various results from

[7], he proved that every continuous function on closed interval is integrable

there. One of the principal advantage of his definition was that it enabled him

to prove the ‘fundamental theorem of calculus’. His definition still applies

to certain class of discontinuous functions. That is, if f : [a, b] → R is a

bounded function which is discontinuous at c ∈ (a, b) and if for all ε > 0,

limε→0

∫ c−ε

a

f(x) dx and limε→0

∫ b

c+ε

f(x) dx

exist, then the definite integral can be defined as∫ b

a

f(x) dx = limε→0

∫ c−ε

a

f(x) dx+ limε→0

∫ b

c+ε

f(x) dx.

Although, the definite integral for a function with any finite number of dis-

continuities in [a, b] can be defined analogously. This approach is not suitable

for functions with an infinite number of discontinuities in [a, b]. Which means

that Cauchy defines the integral mainly for piecewise continuous functions.

In 1854, George Friedrich Bernard Riemann (1826-1866), submitted his

dissertation to the university of Gottingen under the title “Uber die Darstell-

barkeit einer Funktion durch eine trigonometrische Reiche”(On the devel-

opability of a function by a trigonometric series) [40] which was published

posthumously in 1866. In this paper he extends Cauchy’s definition of the

2

integral by recognizing the nonessential nature of the requirement that the

integrand be continuous and generalized Cauchy’s work to include the inte-

grals of bounded functions. He replaced the continuity requirement with the

weaker one that all the sums S(f,P) =∑p

i=0 f(ξi)(xi+1 − xi) converge to a

unique limit.

Riemann himself gave a necessary and sufficient condition for a function

f defined on [a, b] to be Riemann integrable as follows:

Theorem (Riemann, 1854/1866) A necessary and sufficient condition for

the Riemann integrability of a bounded function f over [a, b] is that if ε > 0

and δ > 0, then there exists a partition P : a = x0 < x1 < . . . < xp = b such

that the total length of the subintervals [xk, xk+1], for which the oscillation

ωk is greater than ε, is less than δ.

The oscillation ωk of f over [xk, xk+1] is defined by ωk = Mk −mk, where

Mk = sup {f(x) : xk ≤ x ≤ xk+1} , mk = inf {f(x) : xk ≤ x ≤ xk+1} .

A close observation reveals that the condition in the above theorem is nothing

but the statement that the set of points of discontinuity of the function is to

be of measure zero. Of course, Riemann himself did not regard his condition

in terms of this concept, since it was about 50 years later that such a concept

was explicitly introduced.

Inspired by Riemann’s original proof of integrability, Lebesgue gave the

following characterization of Riemann integrable functions.

Theorem (Lebesgue, 1902) A bounded function f defined on [a, b] is

Riemann integrable if and only if it is continuous almost everywhere.

In the succeeding years, Riemann’s idea was extended in several ways.

Any mathematical operation that involves inverting a derivative can be ad-

dressed with Riemann integral. As an instance, solving differential equations

is often effected with Riemann integrals.

3

However, as the mathematics developed, by the end of the 19th century

it appears that the Riemann theory of integration has some inadequacies,

such as:

i) The class of Riemann integrable functions is quite small. That is, it is

defined only for bounded functions.

For example: Consider f : [0, 1]→ R defined by

f(x) =

0, x = 0

1√x, 0 < x ≤ 1.

This function is not Riemann integrable on [0, 1].

Note that of course we can get around this with improper integrals.

ii) Many functions that are easy to define are not Riemann integrable.

For example: Let E ⊂ [a, b] be such that [a, b] \E is dense in [a, b] and

has measure zero. If f and g are continuous on [a, b] and g 6= 0, define

the function h as follows

h(x) =

f(x), x ∈ E

f(x) + g(x), otherwise.

Clearly, h is discontinuous everywhere hence not Riemann integrable

(The Dirichlet’s function is a special case of this kind).

iii) The Riemann integral does not work well with limiting processes. That

is, limiting operations often lead to insurmountable difficulties. In fact,

if {fn}∞n=1 is a sequence of Riemann integrable functions on [a, b] and

limn→∞

fn(x) = f(x) everywhere in [a, b], then, in general, it is not true

that

limn→∞

∫ b

a

fn(x) dx =

∫ b

a

f(x) dx. (1.1)

Three things may go wrong:

4

a) The limit on the left side of (1.1) may not exist.

b) Even if this limit exists, the function f may not be Riemann in-

tegrable, and then the right side of (1.1) may be meaningless.

For example: Let r1, r2, . . . , rn be n rational numbers in [a, b]. For

each n ∈ N, define fn as follows:

fn(x) =

1, if x = rk, k = 1, 2, . . . , n

0, otherwise.

Observe that each fn is Riemann integrable and∫ bafn(x) dx = 0.

On the other hand, limn→∞

fn(x) = f(x) is the Dirichlet’s function

which is not Riemann integrable.

c) Even if both sides exist, they may not be equal.

For example: For each n ∈ N, define fn as follows

fn(x) = −2n2 x exp(−n2 x2) + 2 (n+ 1)2 x exp(−(n+ 1)2 x2).

Then the fn′s are telescoping and we have

∞∑n=1

fn(x) = −2x exp(−x2).

Now ∫ t

0

[−2x exp(−x2)

]dx = exp(−t2)− 1

and

∞∑n=1

∫ t

0

fn(x) dx =∞∑n=1

[exp(−n2 t2)− exp(−(n+ 1)2 t2)

]= exp(−t2).

Therefore (1.1) is not valid.

5

iv) The Riemann theory of integration is not sufficient in describing a class

of pairs of functions f and F satisfying the fundamental relation:

F (x)− F (a) =

∫ x

a

f(t) dt if and only if F is differentiable and

F′(x) = f(x).

That is, the beauty of the ‘fundamental theorem of calculus’, as exhib-

ited by Cauchy, lost its charm.

For example: A function F : [0, 1]→ R defined as

F (x) =

x2 cos( 1

x2), x 6= 0

0, x = 0

is one such function.

v) The space R([a, b]) of all Riemann integrable functions on [a, b] with

the metric d, called L1-metric, defined as

d(f, g) =

∫ b

a

|f(x)− g(x)| dx

is not complete.

For example: For each n ∈ N, define fn as follows

fn(x) =

0, 0 ≤ x ≤ 1

n+1

n32 [(n+ 1)x− 1], 1

n+1≤ x ≤ 1

n

1√x, 1

n≤ x ≤ 1.

Then {fn} is a Cauchy sequence in C([a, b]) ⊂ R([a, b]) which has no

limit among Riemann integrable functions.

The fact that R([a, b]) is not complete is the most damaging part of the

Riemann integral. The incompleteness of R([a, b]) has consequences which

6

are almost as serious as those which would result from trying to develop real

analysis without the completeness axiom of the real number system. Since

every metric space can be completed, in order to obtain a complete space

generated by C([a, b]), it is necessary to construct a new integral of greater

scope than the Riemann integral.

Many mathematicians contributed in the effort to extend the notion of

the integral in order to remove these drawbacks. Numerous definitions of

the integral for bounded as well as unbounded functions were successively

proposed, the best of which was developed at the very beginning of the 20th

century by the French mathematician Henri Lebesgue (1875-1941). He in-

troduced in his doctoral dissertation at the Sorbonne [30], a notion of the

integral which now bears his name and which overcame most of the short-

comings of the Riemann integral. His integral turns out to be the correct one.

And it is widely used by professional mathematicians. It can integrate un-

bounded, nowhere continuous functions over unbounded domains, and gives

more satisfactory conditions under which one could take limits, or differen-

tiate under the integral sign. Indeed, the Lebesgue theory of integration has

become pre-eminent in contemporary mathematical research, since it enables

one to integrate a much larger class of functions.

There is basic difference between the Riemann theory and the Lebesgue

theory of integration. In Riemann integration, the domain over which the

integral is taken is divided into a number of subintervals, and the integral

is defined as the limit of the Riemann sum for this partition as the length

of the largest subinterval diminishes (tends to zero). On the other hand, in

Lebesgue integration, the domain over which the integral is taken is divided

into a number of measurable sets, and the integral is then defined as the

limit of a certain sum taken for all these measurable sets as the number of

7

measurable sets indefinitely increases. This difference between two methods

leads essentially the distinction that the Riemann integral is based on sub-

dividing the domain of the function (by way of partition) and the Lebesgue

integral is based on subdividing the range of the function.

However, the Lebesgue integration theory also has several inadequacies

such as:

i) Every derivative is not Lebesgue integrable. There exist functions F

that are differentiable on [a, b], but such that F′

is not Lebesgue inte-

grable.

For example: Let F : [0, 1]→ R be defined as

F (x) =

x2 sin( 1

x), x 6= 0

0, x = 0.

Then F′

exists on [0, 1] but F′

is not Lebesgue integrable on [0, 1].

ii) The Lebesgue integral integrates only those functions whose absolute

value is also integrable. That is, Lebesgue integral is absolute integral.

There are some improper integrals which do not exist as Lebesgue in-

tegrals.

For example: The Dirichlet integral∫∞

0sinxxdx.

Indeed, the improper Riemann integral is not a special case of the

Lebesgue integral on R.

iii) For the study of Lebesgue integration, we need considerable prerequi-

sites of measure theory.

iv) And the theory of the Lebesgue integral still considered as a difficult

theory, no matter whether it is based the concept of measure or intro-

duced by other methods.

8

Again, mathematicians were trying to overcome these inadequacies. They

tried to include the class of improper Riemann integrable functions into the

class of Lebesgue integrable functions. But, unfortunately, this is not the

case. Any modification in the definition of Lebesgue integral to admit condi-

tional/nonabsolute convergence will rule out the countable additivity of the

integral for measurable sets.

The problem of reconstruction of original function from its derivative

was first solved by the French mathematician Arnaud Denjoy (1884-1974) in

1912 by introducing an integration process which he called “totalization”[12].

The totalization is rather complicated process that involves the use of trans-

finite numbers. It consisted of essentially three operations, the evaluation of

sequences of Lebesgue integrals, Cauchy extensions of these integrals, and

summing of subsequences of these integrals, the Harnack extension. In 1914,

the same problem was solved independently and very differently [39], based

on certain ideas of de la Vallee Poussin, by the German mathematician Os-

kar Perron (1880-1975). During 1921-1925 H. Hake, P. S. Alexandrov and H.

Looman established the equivalence of the Denjoy and the Perron integrals.

The Denjoy-Perron integral has an additional advantage over the Lebesgue

integral in the sense that it includes the Newton integral and all improper

integrals. The Denjoy and Perron integrals are discussed in [21]. Unfortu-

nately, these new definitions of integral were rather unwieldy and it required

tremendous effort to develop even simple results like integration by parts.

In the following years, investigations into even more general form of in-

tegration continued, mainly because there was lack of good methods to find

primitives of functions. A major breakthrough came about 40 years later

when the Czech mathematician Jaroslav Kurzweil (1926-), in connection

with his research in differential equations [28], defined an integral in 1957

9

using the simple Riemann method with an ingenious modification. He did

not study this integral in detail; just deriving enough of its properties for his

applications. The same method was introduced independently and almost

contemporarily [22] by the Irish mathematician Ralph Henstock (1923-2006)

in the early 1960’s. He made a systematic study of this integral and advanced

it further (although he first considered this concept in 1955). This relatively

recent integral is based on the Riemann theory, which preserves the intuitive

geometrical background of the Riemann integral. It is just as easy to work

with, but with powers to match and in certain cases even exceeds those of

the Lebesgue integral [23, 24, 25]!

While there are a number of ways to express this new theory, the idea of

a gauge function is consistent. Since there are no uniformly accepted titles

for this theory, this new integral goes by several names such as: Henstock

[31, 21], Henstock-Kurzweil [27], Kurzweil-Henstock [29, 32], gauge [46], or

the generalized Riemann integral [33].

The integrals of Denjoy and Perron are in fact equivalent to this integral

[21]. The difference between the Henstock-Kurzweil integral and the Rie-

mann integral is that δ, the gauge, can now be a function rather than a con-

stant [4]. If the function f oscillates wildly near x0 then δ is taken to be small

near x0, and this forces the interval [xi, xi+1] containing x0 to be small. A

function is Riemann integrable if and only if δ can be taken to be a constant

[4]. It is remarkable that the class of Henstock-Kurzweil integrable func-

tions includes the Lebesgue integrable functions [11]. In fact, the Henstock-

Kurzweil integral reduces to the Lebesgue integral whenever we have absolute

integrability [36]. Thus we immediately have all the Lp results. Henstock

has since shown, [25], that the Henstock-Kurzweil integral can encompass in-

tegration over more general sets than Rn and includes Feynman and Wiener

10

integrals, etc. Becasue it has wider applicability and is easier to define than

the Lebesgue integral, there is a movement to replace the Lebesgue integral

with the Henstock-Kurzweil integral in the undergraduate curriculum. The

first formal attempt was made at the Joint Mathematics Meeting in San

Diego, California, in January 1997 in which Robert Bartle, Ralph Henstock,

Jaroslav Kurzweil, Eric Schechter, Stefan Schwabik, and Rudolf Vyborny

distributed an “Open Letter” to the authors of calculus book to include the

Henstock-Kurzweil integral. This letter is available online at the website:

http://www.math.vanderbilt.edu/schectex/ccc/gauge/letter/. How-

ever, it apparently hasn’t had any effect.

One of the important properties of the Lebesgue integral is that the in-

tegrals over unbounded domains or with unbounded integrand are handled

with no special procedure such as must be done with the Riemann integral.

Despite its apparent similarity with the Riemann’s definition, the Henstock-

Kurzweil integral has this same property.

The function sinxx

is integrable over [1,∞]. And, if

f(x) =

x2 cos( 1

x2), for x > 0

0, x = 0,

then∫ 1

0f′

= f(1) for the Henstock-Kurzweil integral but f′

is neither Rie-

mann nor Lebesgue integrable over [0, 1].

The Riemann integral of f′

fails to exist since f′

is not bounded and the

Lebesgue integral does not exist since |f ′ | is not integrable over [0, 1]. Note

that we have the improper Riemann integral

limε→0+

∫ 1

ε

f′(x) dx = lim

ε→0+[f(1)− f(ε)] = f(1).

However, if f′

is changed to be zero on the rationals then this improper

11

Riemann integral no longer exists but the Henstock-Kurzweil integral un-

changed.

The research on gauge integration is getting further deeper and perfect,

which enrich the theory of integration. Contributions by many mathemati-

cians like Bartle [2, 3, 4], Gordon [19, 20, 21], McShane [34, 35], Schwabik

[41, 50], Lee [31, 32], Vyborny [32], Swartz [27, 46], Talvila [47, 48, 49],

Fremlin [17, 18], DiPiazza [15, 14], etc. have focused on various aspects of

this theory in different directions. The integral of Banach space-valued func-

tions has received so much attention that the notion of integrals of real-valued

functions is now relatively becomes classical. The investigations in the study

of gauge type integrals for Banach space-valued functions started around

1990 by the work of Gordon [19]. Since then much of the attention has been

paid to this field and there has been considerable work in this direction for

example [6, 9, 16, 38, 43, 44, 51]. So it remains an active area of research.

T. J. Stieltjes introduced the concept of the integral of one function with

respect to another, known by his name for a special case in a paper [45]

published in 1894. His integral integrates a function f with respect to another

function φ on a given interval.

The present work is mainly concerned with two concepts, Stieltjes-type

integrals and integral transforms. More specifically, an attempt has been

made to extend the Stieltjes integral and generalized Stieltjes integral in

the Henstock-Kurzweil setting to Banach spaces. Also we study the Stieltjes

integral in the Henstock-Kurzweil setting for the Lr-derivative, as introduced

by Calderon and Zygmund [5]. Finally, by the line of Talvila [49], we explore

some integral transforms in context to Henstock-Kurzweil integral.

12

1.2 Mathematical Preliminaries

In this section, we shall survey those definitions and results which are sub-

sequently used throughout the thesis. For integration, we have adopted, and

adapted a recent approach due to Henstock and Kurzweil, called Henstock-

Kurzweil integral. Mostly, we shall take the definitions and results from the

standard books [3] and [21].

Let I = [a, b] be a real closed interval (a < b). A partition of [a, b] will be

denoted by {[xi, xi+1] : 0 ≤ i ≤ p}, where a = x0 < x1 < . . . < xp+1 = b. A

tagged partition P of I is a set of pairs {(ξi, [xi, xi+1])}pi=0 of points ξi and

the closed non-overlapping intervals [xi, xi+1] such that ξi ∈ [xi, xi+1] and

∪pi=0[xi, xi+1] = [a, b]. The points ξi are called the corresponding tag points.

If ∪pi=0[xi, xi+1] ⊂ [a, b], then P is said to be tagged subpartition of I.

A gauge on I is a positive function δ : I → R+. For any given gauge δ on

I, a tagged partition P = {(ξi, [xi, xi+1])}pi=0 is said to be δ-fine if for each i,

[xi, xi+1] ⊆ (ξi − δ(ξi), ξi + δ(ξi)). In this case we write P ≺ δ.

The existence of δ-fine tagged partition of an interval is guaranteed by

the following lemma due to Cousin [10].

Lemma [3]

If δ is a gauge defined on the interval I, then there exists a δ-fine tagged

partition of I.

Riemann Sums [3]

If f : I → R is any function. If P = {(ξi, [xi, xi+1])}pi=0 is any tagged

partition of I, we define the Riemann sum of f corresponding to P as

S(f,P) =

p∑i=0

f(ξi)(xi+1 − xi).

Henstock-Kurzweil Integral [3]

A function f : I → R is said to be Henstock-Kurzweil integrable on I

13

if there exists a real number L ∈ R with the following property: for every

ε > 0 there exists a gauge δ on I such that |S(f,P)− L| < ε whenever P is

a δ-fine tagged partition of I.

We shall denote the space of all Henstock-Kurzweil integrable functions

on I by HK(I). This space is a vector space under the usual operations of

pointwise addition and scalar multiplication. There is a natural seminorm

on HK(I) originally defined by Alexiewicz [1] as follows

A‖f‖ = supt∈[a,b]

∣∣∣∣∫ t

a

f

∣∣∣∣ .Hake’s Theorem [3]

A function f : I → R is in HK(I) if and only if there exists A ∈ R

such that for every c ∈ (a, b), the restriction of f to [a, c] is integrable and

limc→b−

∫ caf = A.

In this case, A =∫ baf .

This result is also true in case of an infinite interval. That is, I = (−∞,+∞).

The Hake’s theorem asserts that there is no such thing as an “improper

integral” for the Henstock-Kurzweil integral.

Multiplier Theorem [3]

Let f ∈ HK(I) and φ ∈ BV(I) be two functions. Then the product

f · φ ∈ HK(I) and∫ b

a

f · φ =

∫ b

a

φ dF = F (b) φ(b)−∫ b

a

F dφ, (1.2)

where F (x) =∫ xaf for x ∈ I and the integrals on right are Riemann-Stieltjes

integrals.

In case of an infinite interval I = [0,∞), the only equation (1.2) changes to∫ ∞a

f · φ = limb→∞

∫ b

a

φ dF = limb→∞

[F (b) φ(b)−

∫ b

a

F dφ

].

14

Dedekind’s Test [46]

Let f, φ : [a,∞) → R be continuous on (a,∞) with limx→a+

φ(x) = 0 and

φ′, absolutely integrable over [a,∞). Assume that F (x) =

∫ xaf , a < x <∞,

is bounded on (a,∞). Then f φ ∈ HK([a,∞)).

Chartier-Dirichlet’s Test [3]

Let f, φ : [a,∞)→ R and suppose that:

i) f ∈ HK([a, c]) for all c ≥ a and F (x) =∫ xaf is bounded on [a,∞).

ii) φ is monotone on [a,∞) and limx→∞

φ(x) = 0.

Then f φ ∈ HK([a,∞)).

Du-Bois Reymond’s Test [3]

Let f, φ : [a,∞)→ R and suppose that:

i) f ∈ HK([a, c]) for all c ≥ a and F (x) =∫ xaf is bounded on [a,∞).

ii) φ is differentiable on [a,∞) and φ′ ∈ L1([a,∞)).

iii) F (x) φ(x) has a limit as x→∞.

Then f φ ∈ HK([a,∞)).

Lemma [49]

Let f ∈ HK(R) and g : R2 → R. Let M denote the class of measurable

subsets of R. For each (A,B) ∈ BV ×M, define the iterated integrals

I1(A,B) =

∫x∈A

∫y∈B

f(x)g(x, y) dy dx,

I2(A,B) =

∫y∈B

∫x∈A

f(x)g(x, y) dx dy.

a) Assume that for each compact interval I ⊂ R, there are constants

MI > 0 and KI > 0 such that∫R VI g(·, y) dy ≤ MI and, for all x ∈ I,

‖g(x, ·)‖1 ≤ KI . If I1 exists on R× R, then I2 exists on BV ×M and

I1 = I2 on BV ×M.

15

b) Assume there exist M,G ∈ L1(R) such that, for almost all y ∈ R,

V g(·, y) ≤M(y) and, for all x ∈ R, |g(x, y)| ≤ G(y). Then I1 = I2 on

BV ×M.

Definition [3]

If E ⊆ I, we say that a function f : I → R belongs to the class

ACδ(E) if for every ε > 0, there exist η > 0 and a gauge δ on E such

that∑p

i=0 |f(xi+1) − f(xi)| < ε whenever P = {(ξi, [xi, xi+1])}pi=0 is a δ-fine

subpartition of E and∑p

i=0 |xi+1 − xi| ≤ η.

We say that f belongs to the class ACGδ(I) if I can be written as a countable

union of sets on each of which the function f is ACδ.

Absolute Continuity [21]

Let f : I → R. We say that f is absolutely continuous on I if for every

ε > 0 there exists η > 0 such that∑p

i=0 |f(xi+1) − f(xi)| < ε whenever

{[xi, xi+1] : 0 ≤ i ≤ p} is a finite collection of non-overlapping intervals in I

that satisfy∑p

i=0(xi+1 − xi) < η. In this case we write f ∈ AC(I).

A sequence {fk} of functions is said to be uniformly absolutely continuous on

I, if for every ε > 0 there exists η > 0 such that∑p

i=0 |fk(xi+1)− fk(xi)| < ε,

∀ k ∈ N, whenever {[xi, xi+1] : 0 ≤ i ≤ p} is a finite collections on non-

overlapping intervals in I that satisfy∑p

i=0(xi+1 − xi) < η.

Lipschitz function [37]

A finite function f : I → R is said to satisfy a Lipschitz condition or is

said to be a Lipschitz function on I if there exists a constant K such that

for any two points x, y ∈ I, |f(x)− f(y)| ≤ K|x− y|.

The constant K is called Lipschitz constant for f .

Regulated function [13]

Let f : I → R and x ∈ I (x 6= b). We say that f has a limit on the right

if limy∈I,y>xy→x

f(x) exists.

16

In this case we write the limit f(x+).

Similarly, we define for each point x ∈ I (x 6= a), the limit on the left of f at

x, which we write f(x−); we also say these limits are one-sided limits of f .

A mapping f : I → R is called regulated function if it has one-sided limits

at every point of I.

Characterization of regulated functions:

A function f : I → R is regulated if and only if there is a sequence

{sn}∞n=1 of step functions on I → R that converges uniformly to f on I.

Bounded variation function [46]

A function f : I → R is said to be of bounded variation on I if V (f, I) =

sup {∑n

i=1 |f(di)− f(ci)|} is finite, where the supremum is taken over all

partitions {[ci, di]}ni=1 of I.

The set of all functions of bounded variation on I is denoted by BV(I).

Let {fk} be a sequence of functions, fk : I → R, ∀ k ∈ N. We say that

{fk} is of uniform bounded variation on I, if there is a constant M such that

|fk| ≤M , and V (fk, I) ≤M , ∀ k ∈ N.

Where V (fk, I) = sup {∑n

i=1 |fk(di)− fk(ci)|}, here the supremum is taken

over all partitions {[ci, di]}ni=1 of I.

Theorem [47]

Suppose f : I → R is Henstock-Kurzweil integrable and {gn} is a sequence

of functions which are of uniform bounded variation on [a, b] and such that

gn → g. Then ∫ b

a

f gn −→∫ b

a

f g as n→∞.

Two-norm convergence [26]

A sequence is said to be convergent in two-norm sense if the sequence is

bounded in one given norm and convergent in another given norm. That is,

a sequence {gk} of functions on I is said to be two-norm convergent to g if

17

V (gk) ≤M for all k and gk is uniformly convergent to g on I.

Mean value theorem [37]

Suppose that the function f : [a, b]→ R is continuous and f : (a, b)→ R

is differentiable. Then there is a point x0 ∈ (a, b) at which f(b) − f(a) =

f′(x0)(b− a).

Banach Space [42]

Let X be a vector space over the field F . A norm on X is a function

‖ · ‖X : X → R satisfying,

i) ‖x‖X ≥ 0 for all x ∈ X. (Non-negativity)

ii) ‖x‖X = 0 if and only if x = 0. (Non-degeneracy)

iii) ‖tx‖X = |t|‖x‖X for all x ∈ X and t ∈ F . (Homogeneity)

iv) ‖x+ y‖X ≤ ‖x‖X + ‖y‖X for all x, y ∈ X. (Triangle inequality)

If ‖ · ‖X is a norm on X, the pair (X, ‖ · ‖X) is called a normed linear space.

A normed linear space is said to be complete if it is complete under the

metric induced by the norm, i.e., if every Cauchy sequence converges. A

complete normed linear space is called Banach space.

Let X be a Banach space. A mapping f : X → R is called functional on

X. The functional on X is said to be linear if for all x, y ∈ X and α, β ∈ F ,

we have f(αx+ βy) = αf(x) + βf(y).

The functional f on X is said to be continuous if for every convergent se-

quence {xn} in X, we have f(xn)→ f(x) whenever xn → x as n→∞.

The collection of all continuous linear functionals on X is called the dual

of X and is denoted by X∗.

Measure, measurable set and measurable functions [21]

A ring of sets is a nonempty class R of sets such that if E,F ∈ R, then

E ∪ F ∈ R and E − F ∈ R.

18

A measure (sometimes also called Lebesgue outer measure) is an extended

real valued, non-negative, and countably additive set function µ, defined on

a ring R, and such that µ(0) = 0. More specifically, if E ⊆ R, then the

measure of E, denoted by µ(E), is defined by

inf

{∞∑k=1

l(Ik) : {Ik} is a sequence of open intervals such that E ⊆ ∪∞k=1Ik

}.

It is obvious that 0 ≤ µ(E) ≤ ∞.

A set E ⊆ R is said to be measurable if for each set A ⊆ R, the equality

µ(A) = µ(A ∩ E) + µ(A ∩ Ec) is satisfied.

A function f : E → R is said to be measurable if E is measurable set and

for each real number r, the set {x ∈ E : f(x) > r} is measurable.

Vitali covering [21]

Let E ⊂ R. A family J of closed, bounded subintervals of R covers E in

the Vitali sense if for every x ∈ E, and ε > 0, there is an interval I ∈ J so

that x ∈ I and µ(I) < ε.

If J covers E in the Vitali sense, we call J a Vitali cover of E.

In other words, the set E is covered by the family J in the Vitali sense if every

point of the set E is contained in arbitrarily small closed intervals belonging

to the family J.

Vitali covering lemma [21]

Let E ⊆ R with µ(E) < ∞. If J is a Vitali cover of E, then for every

ε > 0 there exists a finite collection {Ik : 1 ≤ k ≤ n} of disjoint intervals in

J such that µ (E − ∪nk=1Ik) < ε.

In addition, there exists a sequence {Ik} of disjoint intervals in J such that

µ (E − ∪∞k=1Ik) = 0.

Point of density [21]

Let E be a measurable set and let c be a real number. The density of E

19

at c is defined by

dcE = limh→0+

µ(E ∩ (c− h, c+ h))

2h,

provided the limit exists.

It is clear that 0 ≤ dcE ≤ 1, when it exists. The point c is a point of density

of E if dcE = 1 and a point of dispersion of E if dcE = 0.

A point of density of E is certainly a limit point of E, but not conversely.

For example: Let A = { 1n

: n ∈ N}. Then A has no points of density but has

0 as a limit point.

In general, a set of zero measure has no points of density.

Every real number is a point of density of the set of irrational numbers.

20

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