## ABSTRACT

The goal of this thesis is to extend the notion of integration with respect to

a measure to Lattice spaces. To do so the paper is first summarizing the

notion of integration with respect to a measure on R.

Then, a construction of an integral on Banach spaces called the Bochner

integral is introduced and the main focus which is integration on lattice

spaces is lastly addressed.

Key Words. Banach spaces, Bochner Integral, Integration, Ordered vector

space, Real-valued Mapping Modern Integral, Lattice space, Young-Fatou-

Lebesgue Dominated Convergence Theorem,

** **

## TABLE OF CONTENTS

Certification i

Approval iii

Abstract v

Dedication vii

Acknowledgements ix

General Introduction 1

Chapter 1. Introduction to Integration Theory 5

1.1. Riemann-Stieltjes Integration 5

1.2. Bounded Variation Functions 7

1.3. Lebesgue Integration 11

Chapter 2. Integration with respect to a measure on R : a summary 15

2.1. The construction 15

2.2. Properties of Real-valued Integrable Functions 19

2.3. Spaces of integrable functions 20

Chapter 3. Integration with respect to a measure on Banach spaces

in general 23

3.1. The construction of the integral 23

3.2. The Bochner integral on R 45

i

ii CONTENTS

3.3. Properties and limit theorems for Banach-Valued Bochner

Integral 52

3.4. The space L1(

;A; m;E), in short L1(

;E) 63

3.5. Young-Fatou-Lebesgue Convergence Theorem in L1(

;A; m;E) 72

Chapter 4. Integration of mappings with respect to a measure on

lattice spaces 75

4.1. Another view on the construction of the Bochner integral 75

4.2. Properties of Ordered Vector Spaces 79

4.3. Two main Results of the integration on Ordered Banach

Spaces 81

Chapter 5. Conclusion and Perspectives 83

Bibliography 85

## CHAPTER ONE

Introduction to Integration Theory

1.1. Riemann-Stieltjes Integration

Definition of the Riemann-Stieltjes integral on a compact set

Consider an arbitrary function f : [a; b] ! R.

The Riemann-Stieltjes integral of f on [a; b] associated with F, if it exists,

is denoted by:

I =

Z b

a

f(x) dF(x)

In establishing the existence of the Riemann-Stieltjes integral of a function,

we need the function to be bounded.

Next, we define the Riemann-Stieltjes sums. To do so, for each n 1, we

divide [a; b] into l(n) sub-intervals (l 1).

Let n be a subdivision of [a; b] that divides[a; b] into l(n) sub-intervals.

So,

]a; b] =

l(Xn)?1

i=0

]xi;n; xi+1;n];

where a = x0;n < x1;n < ::: < xl(n);n = b:

5

6 1. INTRODUCTION TO INTEGRATION THEORY

The modulus of the subdivision n is defined by:

m(n) = max

0il(n)?1

(xi+1;n ? xi;n)

Then, in each sub-interval ]xi;n; xi+1;n], we pick an arbitrary point ci;n,

we therefore have the arbitrary sequence (cn)n1 where, cn = (ci;n)1il(n)?1.

we now define a sequence of Riemann-Stieltjes sum associated to the subdivision

n and the vector cn in the form:

(1.1.1) Sn(f; F; a; b; n; cn) =

l(Xn)?1

i=0

f(ci;n)(F(xi+1;n) ? F(xi;n))

in short, Sn(n; cn)

Definition 1.1. A bounded function f is Riemann-Stieltjes integrable

with respect to F if there exists a real number I such that any sequence

of Riemann-Stieltjes sums Sn(n; cn) converges to I as n ! 1 whenever

m(n) ! 0 as n ! 1.

The number I is called the Riemann-Stieltjes integral of f on [a; b]

Now, in particular, if F(x) = x; x 2 R, I is called the Riemann Integral of f

over [a; b] and the sum in formula 1.1.1 is simply called the Riemann Sum.

For the sake of a later use, Let us introduce an important notion called

”‘Bounded Variation Functions”’.

1.2. BOUNDED VARIATION FUNCTIONS 7

1.2. Bounded Variation Functions

Consider a function F : [a; b] ! R.

We define by P(a; b) the class of all partition of the interval [a; b] of the

form:

(1.2.1) = (a = x0 < x1 < ::: < xp = b); p 1

To each 2 P(a; b) represented as in formula 1.2.1, we associate the variation

of F over define by:

VF (; a; b) =

Xp

j=i

jF(xj+1) ? F(xj)j

The total variation of F over [a; b] is defined by:

VF (a; b) = sup

2P(a;b)

VF (; a; b)

Definition 1.2. A function F is said to be of bounded variation if and

only if its total bounded variation over [a; b] is finite, that is:

0 VF (a; b) = sup

2P(a;b)

VF (; a; b)

Example 1.3. (1) Any non-decreasing function F : [a; b] ! R is of

bounded variation.

We have, for all 2 P, VF (; a; b) = F(b) ? F(a), So :

VF (a; b) = F(b) ? F(a) < +1

8 1. INTRODUCTION TO INTEGRATION THEORY

(2) Any non-increasing function F : [a; b] ! R is of bounded variation.

We have, for all 2 P, VF (; a; b) = F(a) ? F(b), So :

VF (a; b) = F(a) ? F(b) < +1

(3) Any continuously differentiable (C1) function F : [a; b] ! R is of

bounded variation.

In fact, since F0 2 C[a; b], then M := sup

x2[a;b]

jF0(x)j < +1 Now, for all

2 P(a; b), by the Mean Value Theorem, 8 j = 1; :::; p; 9 2 [0; 1] such

that:

F(xj) ? F(xj?1) = (xj ? xj?1)F0(xj?1 + j(xj ? xj?1));

So,

VF (; a; b) =

Xp

j=1

(xj ? xj?1)jF0(xj?1 + j(xj ? xj?1))j

M(b ? a)

Therefore,

VF (a; b) = sup

2P

(a; b)VF (; a; b) M(b ? a) < +1

Lemma 1.4. Any bounded variation function on [a; b] is a difference of two

non-decreasing function.

Now, consider a continuous function f : [a; b] ! R. Our interest here is

to show the existence of the Riemann-Stieltjes integral of f. f being so

1.2. BOUNDED VARIATION FUNCTIONS 9

smooth, we should at least expect, for a strong theory of integration, f to

be Riemann-Stieltjes integrable.

However, for what function F can we define the Riemann-Stieltjes integral

of f.

Theorem 1.5. If F is of bounded variation, every continuous function

on [a; b] is integrable, i.e, has a Riemann-Stieltjes integral I denoted by:

I =

Z b

a

f(x) dF(x)

The Riemann-Stieltjes integration is limited. In fact, we started the construction

by first assuming that our function f is bounded and is defined

on the interval of the form [a; b]. Moreover, we also considered different

parameters in establishing the Riemann Sum.

For example, Let F(x) = x. So to determine the Riemann integral of f :

[a; b] ! R, bounded, we need to compute the Riemann Sums. In fact, in

the process of computing the Riemann sums, for a fixed n, we are technically

computing sum of areas of small rectangles of width w = xi?xi?1; 1

i l(n).

However, to approximate the lengths of triangle, we arbitrarily choose a

point ci between xi?1 and xi and we use the image f(ci) of the point ci, in

computing the areas of those triangle. That is, we can choose any ci in

]xi?1; xi].

For our approximation to make sense, we need to have that for any two

points arbitrarily chosen in the sub-interval ]xi?1; xi], the images of those

points are not far from one another in terms of value. In order words, the

10 1. INTRODUCTION TO INTEGRATION THEORY

function f should be continuous.

However, in real-life situation, we hardly meet smooth functions. Therefore,

we make use of the Lebesgue integration which mainly requires only

measurality of functions.

The illustration is given below.

Figure 1. Geometric Interpretation of Riemann integration where we arbitrarily

chose our ci to be xi+1.

1.3. LEBESGUE INTEGRATION 11

1.3. Lebesgue Integration

1.3.1. Distribution function on R.

Definition 1.6. A function F : R ! R is called a distribution function if

and only if:

(i) F is right continuous

(ii) F assigns to intervals non-negative lengths i.e 8 a b, F(b) ? F(a) 0

1.3.2. Lebesgue-Stieltjes measure associated to F. We construct the

Lebesgue-Stieltjes measure on (R; B(R)).

B(R) = (S)

where S = f]a; b]; a < bg is a semi algebra.

Define:

F : S ! R+

]a; b] ! F (]a; b]) = F(b) ? F(a)

F is called the Lebesgue-Stieltjes measure.

If F(x) = x; F = is the Lebesgue measure on R

1.3.3. The Lebesgue-Stieltjes Integral. Let F : R ! R be a distribution

function.

For f, measurable, the Lebesgue-Stieltjes integral of f with respect to the

measure F is denoted as:

I =

Z

f(x) dF (x)

12 1. INTRODUCTION TO INTEGRATION THEORY

The construction of this type of integral, depending on some properties of

f, is given in chapter 3.

In fact, this thesis is mainly about the integration of measurable mappings

with respect to measure.

Also, for the coherence in the theory of integration, it is not a surprise

that the Riemann-Stieltjes integration and the Lebesgue-Stieltjes integration

sometimes coincide.

Example 1.7. (1) Let f : [a; b] ! R, a < b,f bounded.

f is Riemann integrable if and only if f is ?a:e continuous; and the

Riemann and the Lebesgue integrals coincide.

(2) Any Riemann integral on the compact set [a; b] is a Lebesgue integral

on [a; b]

Furthermore the notion of Lebesgue-Stieltjes integration is broader than

the notion of Riemann-Stieltjes integration, because all Riemann-Stieltjes

integrable functions are Lebesgue-Stieltjes integrable but not all Lebesgue-

Stieltjes integrable functions are Riemann integrable.

Example 1.8. f = 1[a;b]

T

Q is Lebesgue integrable but not Riemann integrable.

This chapter is a brief introduction to the theory of integration. All types

of integration have not been discussed. Here, we only introduced the

Riemann-Stieltjes integration and addressed a broader type of integration

1.3. LEBESGUE INTEGRATION 13

called the Lebesgue integration.

In fact, the Lebesgue-Stietjes integration is simply the integration of realvalued

measurable mappings with respect to the Lebesgue-Stieltjes measure.

In coming chapters, we will discuss the integration of measurable functions

with respect to any arbitrary measure on some specific cases. Depending

on the space, we put a finiteness condition on the

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