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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