Integration in Lattice Spaces

Integration in Lattice Spaces

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.

Keywords: 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
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: 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

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

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;

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 measurability 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

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

called the Lebesgue integration.

In fact, the Lebesgue-Stietjes integration is simply the integration of real-valued 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