Sanata Dharma University · ABSTRACT Boby Gunarso. 2016. Gaussian Measures In Hilbert Spaces And...

GAUSSIAN MEASURES IN HILBERT SPACES AND THEIR

APPLICATIONS

BACHELOR THESIS

Presented as Partial Fulfillment of the Requirements

to Obtain the Degree of Sarjana Matematika

Written by:

Boby Gunarso

Student Number: 123114002

MATHEMATICS STUDY PROGRAM

MATHEMATICS DEPARTMENT

FACULTY OF SCIENCE AND TECHNOLOGY

SANATA DHARMA UNIVERSITY

YOGYAKARTA

2016

PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI


APPLICATIONS

BACHELOR THESIS

Presented as Partial Fulfillment of the Requirements

to Obtain the Degree of Sarjana Matematika

Written by:

Boby Gunarso

Student Number: 123114002

MATHEMATICS STUDY PROGRAM

MATHEMATICS DEPARTMENT

FACULTY OF SCIENCE AND TECHNOLOGY

SANATA DHARMA UNIVERSITY

YOGYAKARTA

2016

i


BACHELOR TIESIS


APPLICl|'TIONS

Wrirten b;':

Boby Gunarsl;

Student Numi:er: 1231 l4in2

Approved by:

Thesis Advisor.

I)r.rer.nat. Hen], Pribawanto Suryawan, S.Si.. M.Si Date: 18 July 2016


BACHELOR THESIS


APPLICATIONS

Written by:

Boby Gunarso

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Sudi Mungkasi, S.Si., M.Math.Sc., Ph.D.

iii


STATEMENT OF WORK'S ORIGINALITY

I honestly declare that this thesis, which I have written, does not contain the work or

parts of the work of other people, except those cited in the quotations and the refer-

ences, as a scientific paper should.

Yogyakarta, 1B July 2016

The Writer

[1l[,Boby Gunarso

IV


ABSTRACT

Boby Gunarso. 2016. Gaussian Measures In Hilbert Spaces And Their Appli-

cations. A Thesis. Mathematics Study Program, Departement of Mathematics,

Faculty of Science and Technology, Sanata Dharma University, Yogyakarta.

The Lebesgue measure plays a fundamental role in Rn. It is uniquely determined

(up to some constant) by the properties of being locally finite and invariant under

translation. One may ask a question whether the Lebesgue measure makes sense in

an infinite dimensional space. The answer is negative. In order to build a well de-

fined measure in infinite dimensional spaces, we can incorporate a rapidly decreasing

exponential factor to the Lebesgue measure and hence we obtain the so-called Gaus-

sian measure. Unfortunately, Gaussian measure with identity operator as a covariance

function still cannot be defined in infinite dimensional separable Hilbert spaces. We

have at least 2 ways to remedy this situation. First we can use trace class operator as

a covariance function to show the existence of a Gaussian measure in infinite dimen-

sional separable Hilbert spaces. The second way is by retaining the identity covariance

operator but the consequence is the measure does exist only on a topological dual of

a nuclear space. In this thesis, we will focus only on the first approach, i.e. we con-

struct a Gaussian measure in infinite dimensional separable Hilbert spaces by using a

trace class operator as a covariance function. We start the construction of Gaussian

measure on the real line, on the finite dimensional Euclidean space and finally in an

arbitrary infinite dimensional separable Hilbert space. We also use Gaussian measure

to study Gaussian random variables, white noise mapping, Malliavin derivative, and a

construction of a Brownian motion in a Gaussian Hilbert space.

v


LBMBAR PERNYATAAN PERSETUJUAN

PUBLIKASI KARYA ILMIAH UNTUK KEPENTINGAN AKADEMIS

Yang bertanda tangan di bawah ini, saya mahasiswa Universitas Sanata Dharma:

Nama : Boby Gunarso

NIM :123114002

Demi pengembangan ilmu pengetahuan, saya memberikan kepada Perpustakaan Uni-

versitas Sanata Dharma, karya ilmiah saya yang berjudul:


APPLICATIONS

beserta perangkat yang diperlukan, bila ada. Dengan demikian, saya memberikan

kepada Universitas Sanata Dharma hak untuk menyimpan, mengalihkan ke dalarr ben-

tuk media lain, mengelolanya dalam bentuk pangkalan data, mendistribusikannya se-

cara terbatas, dan mernpublikasikannya di internet atau media lain untuk kepentingan

akademis tanpa perlu meminta izin dari saya maupun mernberikan royalti kepada saya

selama tetap mencantumkan nama saya sebagai penulis.

Dernikian pernyataan ini saya buat dengan sebenarnya.

Dibr-rat cli Yo-eyakarta

Pada tarr,ggal 25 A-er"rstus 2016

Yan-t menyatakan,

i, ',

,,

J'l ,\l'\r"ry

Boby Gunarso


ACKNOWLEDGEMENTS

First of all, I would like to thank my Lord, Jesus Christ, for the blessing so that I can

finally finish my bachelor thesis. I thank for all the ways, strengths, and courage that

He gave me during the process of finishing this thesis.

During the writing of this thesis, I recieve supports and assistance from many peo-

ple. Therefore, I would like to thank especially to:

1. Dr.rer.nat. Herry Pribawanto Suryawan, M.Si., as my thesis advisor, for his

guidance, suggestion, and correction during the writing of this thesis.

2. Y.G. Hartono, S.Si., M.Sc., Ph.D, as my academic advisor, for his guidance and

support during my bachelor study.

3. Prof. Dr. Frans Susilo, SJ and Prof. Dr. Christiana Rini Indrati, M.Si., as the

examiners, for their valuable correction and suggestions of this thesis.

4. All lecturers of the mathematics study program, for all the knowledge and sup-

port given to me during my bachelor study.

5. My family, for their support during my bachelor study.

6. All my friends in mathematics study program, for sharing happiness, support,

and knowledge during our study.

Finally, I realize that this thesis is still far from perfect. I welcome any critics and

suggestions to make this thesis better. I hope this thesis will be useful for everyone

who would like to read and have interest in this thesis topic.

Yogyakarta, 18 July 2016

The Writer

vii


TABLE OF CONTENTS

TITLE PAGE i

APPROVAL PAGE ii

ENDORSEMENT PAGE iii

STATEMENTS OF WORK’S ORIGINALITY iv

ABSTRACT v

PERNYATAAN PERSETUJUAN PUBLIKASI KARYA ILMIAH vi

ACKNOWLEDGEMENTS vii

TABLE OF CONTENTS viii

CHAPTER 1 INTRODUCTION 1

A. Research Background . . . . . . . . . . . . . . . . . . . . . . . . . . 1

B. Research Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

C. Problem Limitation . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

D. Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

E. Research Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

F. Research Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

G. Systematics Writing . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

CHAPTER 2 TOOLS FROM MEASURE THEORY AND FUNCTIONAL

ANALYSIS 5

A. Tools from Measure Theory . . . . . . . . . . . . . . . . . . . . . . . 6

B. Tools from Functional Analysis . . . . . . . . . . . . . . . . . . . . . 20

C. Events and Random Variables . . . . . . . . . . . . . . . . . . . . . . 38

viii


D. Motivation to Measure in Infinite Dimensional Spaces . . . . . . . . . 41

CHAPTER 3 GAUSSIAN MEASURES IN HILBERT SPACES 46

A. Mean and Covariance of Probability Measures in Hilbert Spaces . . . 46

B. Law of a Random Variable . . . . . . . . . . . . . . . . . . . . . . . 48

C. Gaussian Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1. Gaussian Measures in R . . . . . . . . . . . . . . . . . . . . 51

2. Gaussian Measures in Rn . . . . . . . . . . . . . . . . . . . . 52

3. Gaussian Measures in Hilbert Spaces . . . . . . . . . . . . . . 54

CHAPTER 4 APPLICATIONS OF GAUSSIAN MEASURES 60

A. Random Variables on a Gaussian Hilbert Spaces . . . . . . . . . . . . 60

B. Linear Random Variables on a Gaussian Hilbert Spaces . . . . . . . . 63

C. Equivalence Classes of Random Variables . . . . . . . . . . . . . . . 64

D. The Cameron Martin Space and The White Noise Mapping . . . . . . 67

E. The Malliavin Derivative . . . . . . . . . . . . . . . . . . . . . . . . 73

F. Approximation by Exponential Functions . . . . . . . . . . . . . . . 74

G. The Malliavin-Sobolev Space D1,2pH ,µq . . . . . . . . . . . . . . . 76

H. Brownian Motion in Gaussian Hilbert Space . . . . . . . . . . . . . . 79

CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS 85

A. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

B. Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

BIBLIOGRAPHY 87

ix


CHAPTER 1

INTRODUCTION

A. Research Background

In Euclidean geometry we need the concept of a measure to know the measure µpSq of

a solid body S. The Lebesgue measure generalizes the concept of the usual measure in

Euclidean geometry. In one, two, and three dimensional Euclidean spaces, we refer to

this measure as length, area, and volume of S, respectively. The classical idea to build

measure in Euclidean space is by partitioning the body we measured into a finitely

many components and then applying some rigid motions into those components to

form a simpler body which presumably has the same measure. After the presence of

analytic geometry, Euclidean geometry became interpreted as the study of Cartesian

product Rn of the real line R. Using this analytic foundation rather than the classical

geometrical one, it is no longer intuitively obvious how to measure any subsets of Rn.

This is what we learn in the theory of measure, since we want to keep some nice prop-

erties of a measure (e.g. invariant under isometries), so we restrict ourselves to only

measure some nice subsets of Rn (i.e. σ -algebra) instead of all subsets of Rn. In the

real line, the existence of such non-measurable set can be seen as follows: since Q is

an Abelian additive subgroup of R, then the quotient group RQ form a partition of R

into disjoint cosets. Next, by the denseness of each coset A P RQ, we can have the

following set of all coset representatives V txA : xA P AXr0,1s and A P RQu called

Vitali set. Using the countable additivity and translation invariant properties of a mea-

sure, it is easy to show that this set is indeed a non-measurable set. Unfortunately, not

all measures we have in finite dimensional space can be extended in a naive way to

the infinite dimensional spaces. As we will discuss in the next chapter that the only

locally finite and translation invariant Borel measure µ on an infinite dimensional sep-

arable Banach space is the trivial measure. So we will introduce the so-called Gaussian

1


measure in infinite dimensional spaces. In this thesis we also give an overview to the

above discussed problem on measures in infinite dimensional spaces. First we will

recall some important concepts from measure theory and functional analysis and then

we shall construct the Gaussian measures in the finite and infinite dimensional spaces.

B. Research Problems

There are several problems to be answered in this research:

1. What is the motivation behind measure in infinite dimensional spaces?

2. How to construct a Gaussian measure in infinite dimensional separable Hilbert

spaces?

3. How to define random variables in Gaussian Hilbert spaces and what are their

properties?

4. How to use Gaussian measure to define and study the Cameron-Martin space

and white noise mapping?

5. How to formulate the Malliavin derivative of functions in Malliavin-Sobolev

space?

6. How to construct Brownian motion B Bt , t P r0,T s, in a Gaussian Hilbert

space?

C. Problem limitation

There are two limitations in this research as follows:

1. The state space of the Gaussian measure discussed in this thesis is limited to

infinite dimensional separable Hilbert space.

2


2. The applications of the Gaussian measure discussed in this thesis are limited

to the use of Gaussian measure to study random variables on a Gaussian Hilbert

space, the Cameron-Martin space and white noise mapping, the Malliavin deriva-

tive, and construction of a Brownian motion in a Gaussian Hilbert space.

D. Research Objectives

The objective of this research is to understand the theoretical background of Gaussian

measures in infinite dimensional Hilbert spaces and some of their applications.

E. Research Benefits

The benefits of this research are the writer can obtain more knowledge about measure

in infinite dimensional spaces and its applications and also as a reference for further

study on this topics.

F. Research Methods

The method of this research is by literature review method, that is, by reading some

books and papers related to the topic of this research.

G. Systematics Writing

CHAPTER I. INTRODUCTION

A. Research Background

B. Research Problems

C. Problem Limitation

D. Research Objectives

3


E. Research Benefits

F. Research Methods

G. Systematics Writing

CHAPTER II. TOOLS FROM MEASURE THEORY AND FUNCTIONAL ANALY-

SIS

A. Tools from Measure Theory

B. Tools from Functional Analysis

C. Events and Random Variables

D. Motivation to Measure in Infinite Dimensional Spaces

CHAPTER III. GAUSSIAN MEASURES IN HILBERT SPACES

A. Mean and Covariance of Measures in Hilbert Spaces

B. Law of a Random Variable

C. Gaussian Measures

CHAPTER IV. APPLICATIONS OF GAUSSIAN MEASURES

A. Random Variables on a Gaussian Hilbert Spaces

B. Linear Random Variables on a Gaussian Hilbert Spaces

C. Equivalence Classes of Random Variables

D. The Cameron Martin Space and The White Noise Mapping

E. The Malliavin Derivative

F. Approximation by Exponential Functions

4


G. The Malliavin-Sobolev Space D1,2pH ,µq

H. Brownian Motion in Gaussian Hilbert Space

CHAPTER V. CONCLUSIONS AND RECOMMENDATIONS

A. Conclusions

B. Recommendations

BIBLIOGRAPHY

5


CHAPTER 2

TOOLS FROM MEASURE THEORY AND FUNCTIONAL

ANALYSIS

In this chapter we will discuss some important concepts from measure theory and

functional analysis that play important role in the next chapters.

A. Tools from Measure Theory

Definition 2.1.1 (Sigma Algebra). Let Ω be a nonempty set, we call F 2Ω a σ -

algebra on Ω if it satisfies the following axioms

(i) (Empty set) H PF .

(ii) (Complement) If A PF , then Ac PF .

(iii) (Countable unions) If A1, ...,An, ... PF , then8

n1 An PF .

We also call the ordered pair pΩ,F q as a measurable space and A PF as a measur-

able set.

This definition shows that a σ -algebra F is a countable version of a (concrete) Boolean

algebra on Ω. From these axioms, it is clear that the trivial algebra tH,Ωu is the

smallest σ -algebra and the discrete σ - algebra 2Ω is the largest σ -algebra on Ω. We

can also deduce from the generalized de Morgan’s law that F is closed under countable

intersections.

It is easy to see that the intersection (finite, countably infinite, or even uncountable)

of arbitrary σ -algebras on Ω is again a σ -algebra on Ω. Therefore, we can obtain the

following definition.

Definition 2.1.2 (Generation of σ -algebras). Let Ω be a nonempty set and G 2Ω.

We define σpG q to be the intersection of all the σ -algebras on Ω that contain G and

6


we call it the σ -algebra generated by G . Clearly, σpG q is the smallest σ -algebra on

Ω containing G . In particular, we call the σ -algebra generated by all open subsets of

Ω, denoted by BpΩq, as the Borel σ -algebra on Ω, i.e., BpΩq σpτq where τ is a

topology on Ω.

From the definition above, we can directly show that if PpAq=”the set A has prop-

erty P” and satisfies the following axioms

(i) PpAq is true for every A P G .

(ii) PpHq is true.

(iii) If PpAq is true for some A Ω, then PpAcq is also true.

(iv) If An Ω and PpAnq is true for every n P N, then PpnPNAnq is also true,

then PpAq is true for every A P σpG q.

Definition 2.1.3 (Dynkin’s π λ system). Let Ω be a nonempty set and A ,B be

collections of subsets of Ω. We call A a π-system if A1XA2 PA for every A1,A2 PA .

We call B a λ -system if it satisfies the following axioms

(i) (Empty set) H PB.

(ii) (Complement) If B PB, then Bc PB.

(iii) (Countable disjoint unions) If B1, ...,Bn, ... PB and if BiXB j H for i j, then8n1 Bn PB.

Clearly a λ -system which is also a π-system is a σ -algebra since8

n1 Bn B1 YpB2zB1q Y pB3zB2zB1q Y ... P B for any sequence pBnqnPN in B. Moreover, every

λ -system B is closed under proper differences, i.e., if B1,B2 PB with B2 B1, then

B1zB2 PB. Now we will prove the following lemma before stating the Dynkin’s πλ

theorem.

7


Lemma 2.1.1. Let A be a π-system of Ω and lpA q be the smallest λ -system contain-

ing A as a subset. Then, lpA q is a σ -algebra.

Proof. It suffices to prove that lpA q is a π-system. Set lA tB Ω : AXB P lpA qufor any A P lpA q. It is clear that H P lA and lA is closed under countable disjoint

unions. Next, if B P lA, then AXBc AzpAXBq is a proper difference of sets in the

λ -system lpA q, and so is in lA. This shows that lA is a λ -system for any A P lpA q.Now suppose A PA . Then AXB PA lpA q for every B in the π-system A . Hence,

B P lA and thus lA A . But since lpA q is the smallest λ -system containing A , then

we have lA lpA q. Therefore, AXB P lpA q for every B P lpA q. Finally, let B P lpA qand consider lB tA Ω : AXB P lpA qu. Using our result before, we conclude that

lB is a λ -system and lB A . But this means AXB P lpA q for every A P lpA q. Since

B P lpA q is arbitrary, then the result follows.

Theorem 2.1.1 (Dynkin’s π λ theorem). Let A be a π-system of subsets of Ω and

B a λ -system of subsets of Ω such that A B. Then, σpA q B.

Proof. Let Ω be a nonempty set and A ,B be π-system and λ -system of Ω, respec-

tively, such that A B. Then, by definition of lpA q as the smallest λ -system con-

taining A , we have lpA q B. From the previous lemma, we know that lpA q is a

σ -algebra, and since A lpA q, we have σpA q lpA q. This completes the proof,

since lpA q B.

Definition 2.1.4 (Measure). Let pΩ,F q be a measurable space. If µ : F ÑR, where

R is the extended nonnegative real number line R r0,8s, satisfies the following

axioms

(i) (Null empty set) µpHq 0

(ii) (Countable additivity) If A1, ...,An, ... is a sequence of pairwise disjoint sets in

F , then µp8n1 Anq

°8n1 µpAnq,

8


then we call µ a measure on pΩ,F q and pΩ,F ,µq a measure space. If µpΩq 1,

then we call µ a probability measure and pΩ,F ,µq a probability space.

Definition 2.1.5 (Outer Measure). Let Ω be an arbitrary set. A set function µ : 2Ω ÑR is called an outer measure on Ω if it satisfies the following axioms

(i) (Null empty set) µpHq 0.

(ii) (Monotonicity) If A1,A2 P 2Ω such that A1 A2, then µpA1q ¤ µpA2q.

(iii) (Countable subadditivity) If A1, ...,An, ... P 2Ω, then µp8n1 Anq ¤

°8n1 µpAnq.

From the third axiom, an outer measure µ is countably subadditive on the σ -algebra

2Ω. If it is also additive on Ω, i.e., µpA1YA2q µpA1qµpA2q for every A1,A2 P2Ω such that A1XA2 H, then we have µp8

n1 Anq ¥ µpNn1 Anq

°Nn1 µpAnq

for every N P N and pairwise disjoint sets A1, ...,An, ... P 2Ω so that µp8n1 Anq ¥°8

n1 µpAnq. Using this result together with the countable subadditivity of µ, we

conclude that µ is countably additive on 2Ω so that it is a measure on the σ -algebra

2Ω. The above definition also allow us to define the concept of measurability. A subset

A of Ω is said to be measurable with respect to µ (or µ-measurable) if it satisfies the

following Caratheodory condition

µpT q µ

pT XAqµpT XAcq for any set T P 2Ω.

Definition 2.1.6 (Covering class of a set). Let Ω be an arbitrary set and A 2Ω be a

collection of subsets of Ω such that

(i) (Empty set) H PA .

(ii) (Countable covering) Ω has a countable covering in A , i.e., there exist a se-

quence A1, ...,An, ... PA such that8

n1 An Ω,

then we call A as the covering class of Ω

9


Clearly, the first axiom implies that every subset A of Ω has a countable covering, i.e.,

there exist a collection of sets A1, ...,An, ... PA such that8

n1 An A.

Theorem 2.1.2. Let Ω be an arbitrary set and A be a covering class of Ω. Let δ be

an arbitrary set function δ : A Ñ r0,8s such that δ pHq 0. Then, the set function

µ : 2Ω Ñ r0,8s defined by

µpAq inf

# 8

n1

δ pAnq : A1, ...,An, ... PA and8¤

n1

An A

+

is an outer measure on Ω based on δ .

Proof. It is obvious that µpAq P r0,8s for every A P 2Ω and µpHq ¤ δ pHq 0

and hence µpHq 0. Next, if A1,A2 P 2Ω such that A1 A2, then every covering

sequence in A for A2 is also a covering sequence for A1. Let M and N be two sets

containing all numbers in R on which we take the infimum to obtain µpA1q and

µpA2q, respectively. Then, MN and consequently µpA1q infM¤ infN µpA2qwhich proves the monotonicity of µ. Last, let A1, ...,An P 2Ω. Given ε ¡ 0, then

for each n P N there exist a sequence pAn,kqkPN in A such that8

k1 An,k An and°8k1 δ pAn,kq ¤ µpAnq ε

2n . Therefore,8

n1p8

k1 An,kq 8

n1 An which implies

that

µp

8¤n1

Anq ¤8

n1

r8

k1

δ pAn,kqs ¤8

n1

pµpAnq ε

2n q 8

n1

µpAnq ε

and since ε ¡ 0 is arbitrary, the conclusion follows.

Definition 2.1.7 (Lebesgue measure). Let I0 be the collection of H and all open inter-

vals in R. Let δ be a set function on I0, i.e., δ : I0 Ñ r0,8s such that δ pHq 0 and

for any sequence pInqnPN of disjoint intervals in R, we have δ p8n1 Inq

°8n1 δ pInq.

If I is an interval in R with endpoints a,b PR where a b, then we define δ pIq ba

and if I is an infinite interval in R, we define δ pIq 8. Define the Lebesgue outer

10


measure on R, µ : 2R Ñ r0,8s, as follows

µpAq inf

# 8

n1

δ pAnq : A1, ...,An, ... P I0 and8¤

n1

An A

+

We call the restriction of µ to the collection of all Lebesgue measurable (i.e. mea-

surable with respect to µ) sets in R as the Lebesgue measure on R. We also call the

corresponding measure space as the Lebesgue measure space.

Lemma 2.1.2. For every countable subset A of R, we have µpAq 0.

Proof. First we will show that every singleton has measure zero. Let x P R and ε ¡0. Then px ε

2 ,x ε

2q is an open cover of txu. Hence, we have µptxuq ¤ δ ppxε

2 ,x ε

2qq ε . Since ε ¡ 0 is arbitrary, it follows that µptxuq 0. Now let A be a

countable subset of R, then A nPN txnu is a countable union of null sets An txnu

and therefore µpAq ¤°nPN µpAnq 0 and the conclusion follows.

Theorem 2.1.3. µ δ on the set of all intervals in R.

Proof. First, consider the finite closed interval I ra,bs. Let ε ¡ 0, then I ra,bs pa ε

2 ,b ε

2q P I0, i.e., pa ε

2 ,b ε

2q(

is an open cover in I0 of I. Hence, we have

µpIq ¤ δ ppa ε

2 ,b ε

2qq δ pIq ε . Since ε is arbitrary, then µpIq ¤ δ pIq. On the

other side, let pInqnPN I0 be a covering sequence of I. We will show that°

nPN δ pInq ¥δ pIq. If there exist k such that Ik P pInqnPN is an infinite interval, then

°nPN δ pInq ¥

δ pIkq 8 ¡ δ pIq and we are done. So we assume that each In is a finite open

interval. Let pJnqnPN be a subsequence of pInqnPN such that JnX I H for every n and

Jk Jl if k l. Then°

nPN δ pJnq ¤°

nPN δ pInq. Since I is compact and pJnqnPN is

an open cover of I, then there exist a finite subcover pJnkqNk1 pJnqnPN of I. Assume

that Jni pani,bniq with an1 ¤ ...¤ anN . Obviously, if ani an j for i j, then we have

Jni Jn j or vice versa which contradicts the fact that Jk Jl if k l. Therefore, we

have an1 ... anN . Next we will show that am1 bm for every m P tn1, ...,nN1u.

11


If not, then am1 ¥ bm for some m. Since Jn X I H for every n, then there exist

xm P Jm X I and xm1 P Jm1 X I such that am xm bm ¤ am1 xm1 bm1.

Since I is an interval and xm,xm1 P I, we have rxm,xm1s I. Consider the following

two possibilities. If bm am1, then bm P I but bm R Ni1 Jni . If bm am1, then

rbm,am1s I but rbm,am1s N

i1 Jni which contradicts the fact that pJniqNi1 is an

open cover of I. Thus we have am1 bm for every m P tn1, ...,nNu. So we have the

following inequalities

¸nPN

δ pInq ¥¸nPN

δ pJnq ¥N

i1

δ pJniq pbn1 an1q ...pbnN1 anN1qpbnN anN q

¥ pan2 an1q ...panN anN1qpbnN anN q

¥ bnN an1

¥ ba

δ pIq,

i.e., δ pIq is a lower bound of°

nPN δ pInq so that µpIq ¥ δ pIq and the conclusion

follows. Second, let I pa,bq be a finite open interval. Using the previous result and

the monotonicity and subadditivity of µ as an outer measure, we have the following

inequalities

µppa,bqq ¤ µ

pra,bsq ¤ µptauqµ

ppa,bqqµptbuq µ

ppa,bqq

which implies that µppa,bqq µpra,bsq δ pra,bsq δ ppa,bqq. Third, if I is a finite

interval of the form I pa,bs, then we have µppa,bqq ¤ µppa,bsq ¤ µppa,bqq µptbuq µppa,bqq so that µppa,bsq µppa,bqq δ ppa,bqq δ ppa,bsq. Clearly

for I ra,bq we also have µpra,bqq δ pra,bqq. Last, if I is an infinite interval, say

of the form I pa,8q, then we have µppa,8qq ¥ µppa,nqq δ ppa,nqq n a

for any n ¡ a so that µppa,8qq 8 δ ppa,8qq. Similar arguments also hold

12


for any other infinite interval I.

Definition 2.1.8 (Translation Invariance). Let Ω be a vector space, pΩ,F ,µq a mea-

sure space, and A x ta x : a P Au for any A PF and x P Ω. Then

(i) The σ -algebra F is called invariant under translation if for every measurable

set A and x P Ω, A x is also measurable.

(ii) The measure µ is called invariant under translation if F is invariant under

translation and for every measurable set A and x PΩ, Ax has the same measure

as A itself.

(iii) pΩ,F ,µq is called a translation invariant measure space if both F and µ are

invariant under translation.

Lemma 2.1.3 (Translation invariance of the Lebesgue outer measure). Lebesgue outer

measure µ on R is invariant under translation.

Proof. Clearly, the σ -algebra 2R is invariant under translation. Let A P 2R and x P R.

Take an arbitrary open interval covering pInqnPN of A in I0, then we have8

n1pIn xq p8

n1 Inq x A and hence°8

n1 δ pInq °8

n1 δ pIn xq ¥ µpA xq. Since

pInqnPN is an arbitrary open interval cover of A, then µpAq ¥ µpA xq from the

definition of µ as the infimum of°8

n1 δ pInq. Conversely, by applying the similiar

method to the set Ax and its translate pAxqx A, we have the reverse inequality

µpA xq ¥ µpAq and the conclusion follows.

Theorem 2.1.4 (Translation invariance of the Lebesgue measure space). The Lebesgue

measure space is translation invariant.

Proof. Let A be a Lebesgue measurable set and x P R, then for any subset T of R, we

have

µpT XpA xqqµ

pT XpA xqcq µptT XpA xqu xqµ

ptT XpA xqcu xq

13


µppT xqXAqµ

ppT xqXAcq

µpT xq

µpT q

by using Lemma 2.1.2 and the fact that A is measurable. This shows that A x also

satisfies the Caratheodory condition and therefore it is Lebesgue measurable. By using

this result together with lemma 2.1.2, we also conclude that

µpA xq µpA xq µ

pAq µpAq,

i.e., the Lebesgue measure µ is translation invariant.

Definition 2.1.9 (Complete measure space, complete extension, and completion of a

measure space). A measure space pΩ,F ,µq is called a complete measure space if

every subset A0 of a null set (i.e. a set with measure zero) A P F has measure zero.

If there exist a complete measure space pΩ,F0,µ0q such that F0 F and µ0 µ

on F , then we say that pΩ,F0,µ0q is a complete extension of pΩ,F ,µq. If it is the

smallest complete extension of pΩ,F ,µq, i.e, if for any complete extension pΩ,F1,µ1qof pΩ,F ,µq we have F1 F0 and µ1 µ0 on F0, then we call it the completion of

the corresponding measure space.

Theorem 2.1.5. The Lebesgue measure space is complete.

Proof. First, notice that if A P 2R has Lebesgue outer measure 0, then it is Lebesgue

measurable. Clearly, for any testing set T P 2R, we have 0 ¤ µpT XAq ¤ µpAq 0

and hence µpT XAq 0. Therefore, µpT q ¤ µpT XAcq µpT XAqµpT XAcqso that A is Lebesgue measurable. Next, assume that A has Lebesgue measure zero and

B A, then µpBq ¤ µpAq µpAq 0 so that B is Lebesgue measurable.

Since every interval is Lebesgue measurable and every open set is a union of count-

14


able open intervals, it follows that every Borel set is Lebesgue measurable. The Borel

measure space on R is obtained by restricting the domain of the Lebesgue measure µ

from the Lebesgue σ -algebra to the Borel σ -algebra on R. We call such a measure as

Borel measure on R and denote it by µB. The previous theorem then implies that the

Lebesgue measure space is a complete extension of the Borel measure space.

Definition 2.1.10 (Measurable function). Let pX ,A q and pY,Bq be measurable spaces.

A function f : X Ñ Y is called measurable (BA -measurable) if for all B P B, we

have f1pBq PA .

The following theorem simplifies our problem of checking the measurability of a

function from the entire σ -algebra B to only its generator.

Theorem 2.1.6. Let pX ,A q, pY,Bq be measurable spaces and G 2Y be the generator

of B, i.e., B σpG q. Then a function f : X ÑY is measurable if and only if f1pGq PA for every G P G .

Proof. Clearly if f is measurable, then f1pGq P A for every G P G . Define M M PB : f1pMq PA

(. It is easy to show that M is a σ -algebra as follows: since

f1pY q X PA , then Y PM . If M PM , then f1pMq PA , and hence p f1pMqqc f1pMcq P A so that Mc P M . Finally, if M1, ...,Mn, ... P M , then f1pMiq P A for

every i P N which implies thatn

i1 f1pMiq f1pni1pMiqq PA . Therefore, since

G M , then σpG q σpM q, i.e, B M and the conclusion follows.

Definition 2.1.11 (Simple function). A function is called a simple function if its range

is a finite set.

An R-valued simple function φ always has a representation φ °nk1 ak1Ek where

ak P R and Ek φ1ptakuq. This definition of simple functions play a fundamental

role in measure theory. This is also an implication of the following theorem.

15


Theorem 2.1.7 (Approximation by simple functions). If f : X Ñ R is a nonnegative

measurable function, then there is a monotone increasing sequence of nonnegative

simple functions ϕn : X Ñ R such that ϕn Ñ f pointwise as n Ñ8.

Proof. For each n PN, we divide the interval r0,2nq in the range of f into 22n subinter-

vals of width 2n as follows. Define Ik,n rk2n,pk 1q2nq for k 0,1, ...,22n 1

and In r2n,8s. We also define Ek,n f1pIk,nq and En f1pInq. Then, it is easy to

see that the sequence of simple functions pϕnqnPN where ϕn °2n1

k0 k2n1Ek,n 2n1En

satisfies the required properties.

Note that every function f can be written as a sum of two positive functions,

i.e., f f f where f maxt f ,0u and called the positive part of f and f maxt f ,0u and called the negative part of f . This means that we can prove that some

properties hold for f just simply by proving that it is true for the corresponding simple

functions.

Definition 2.1.12 (Lebesgue Integral). Let pΩ,F ,µq be a measure space. The Lebesgue

integral over Ω of a measurable simple function φ : Ω Ñ R is defined as

»Ω

φdµ »

Ω

n

k1

ak1Ekdµ n

k1

akµpEkq.

If f : Ω Ñ R is a nonnegative measurable function, then the Lebesgue integral of f

over Ω is defined as³

Ωf dµ sup

³Ω

φdµ : φ is simple and 0 ¤ φ ¤ f(

. Finally, for

any measurable function f : ΩÑRRYt8,8u, the Lebesgue integral of f over

Ω is defined as³

Ωf dµ ³

Ωfdµ ³

Ωfdµ .

Theorem 2.1.8 (Monotone Convergence Theorem). Let p fnqnPN be a sequence of non-

negative measurable functions in a measure space pΩ,F ,µq which increasing point-

wise to f , then »Ω

f dµ »

Ω

limnÑ8 fndµ lim

nÑ8

»Ω

fndµ.

16


Proof. Since fn is measurable and increasing pointwise to f , then f is measurable.

Also, since fn ¤ f , then³

Ωfndµ ¤ ³

Ωf dµ . Take any c P p0,1q and fixed a simple

function 0 ¤ ϕ ¤ f . Let An tx P Ω : fnpxq ¥ cϕpxqu. It is clear that An is increasing.

To show that X nPNAn, let x P Ω. Then

(i) If ϕpxq 0, then x P An for any n P N.

(ii) If ϕpxq ¡ 0, from the fact that f pxq ¥ ϕpxq ¡ cϕpxq, then there exist n P N such

that fnpxq ¥ tϕpxq.

This means that x P An. For any A P F , set vpAq ³A cϕdµ c

°mn1 ynµpAXEnq

where ϕ °mn1 yn1En and yn ¥ 0. It is easy to check that v is a measure on Ω. Fur-

thermore, from the continuity of a measure and monotonicity of the Lebesgue integral,

we have

»Ω

cϕdµ vpΩq vp¤nPN

Anq limnÑ8vpAnq lim

nÑ8

»An

cϕdµ

¤ limnÑ8

»An

fndµ

¤ limnÑ8

»Ω

fndµ.

That is, c³

Ωϕdµ ¤ limnÑ8

³Ω

fndµ . Since c P p0,1q is arbitrary, we have³

Ωϕdµ ¤

limnÑ8³

Ωfndµ . Finally by taking the supremum of ϕ in the last inequality we obtain

the desired result.

Definition 2.1.13 (Integral of an almost everywhere defined function). Let pΩ,F ,µqbe a measure space and f : AzN ÑR be a measurable function on AzN where A,N PF ,A N, and N is a null set. We write

³A f dµ for

³Arf dµ where rf is a nonnegative

17


extended real valued measurable function defined on A by setting

rf pxq $''&''%

f pxq if x P AzN

0 if x P N.

It is a fact that³

Arf dµ ³

AzN rf dµ ³Nrf dµ ³

AzN rf dµ .

The definition above of³

A f dµ for a function f which is defined only a.e. (almost

everywhere) on A is for convenience of not having to write³

AzN f dµ .

Theorem 2.1.9 (Fatou’s lemma). Let pΩ,F ,µq be a measure space and p fnqnPN be an

arbitrary sequence of nonnegative extended real valued measurable functions on a set

A PF , then we have »A

liminfnÑ8 fndµ ¤ liminf

nÑ8

»A

fndµ.

In particular, if f limnÑ8 fn exist a.e. on A, then

»A

f dµ ¤ liminfnÑ8

»A

fndµ.

Proof. By definition, liminfnÑ8 fn limnÑ8 infk¥n fk. Clearly, pinfk¥n fkqnPN is

increasing and hence by the monotone convergence theorem

»A

liminfnÑ8 fndµ

»A

limnÑ8 inf

k¥nfkdµ lim

nÑ8

»A

infk¥n

fkdµ

liminfnÑ8

»A

infk¥n

fkdµ

¤ liminfnÑ8

»A

fndµ

Last, if f limnÑ8 fn exist a.e. on A, then f liminfnÑ8 fn a.e. on A. Thus the

conclusion follows by Definition 2.12.

Theorem 2.1.10 (Lebesgue dominated convergence theorem). Let pΩ,F ,µq be a mea-

18


sure space and fn : X Ñ R be a sequence of measurable functions such that fn Ñ f

pointwise as nÑ8. Suppose that there exist an integrable function g such that | fn| ¤g for every n P N. Then fn and f are also integrable and limnÑ8

³Ω| fn f |dµ 0.

Proof. Obviously fn and f are integrable. Also, 2g| fn f | is nonnegative and mea-

surable. By Fatou’s lemma

»Ω

liminfnÑ8p2g| fn f |qdµ ¤ liminf

nÑ8

»Ω

p2g| fn f |qdµ

Since fn Ñ f , the left hand side of the inequality above is just³

Ω2gdµ and the right

hand side is equal to

liminfnÑ8

»Ω

2gdµ »

Ω

| fn f |dµ

»

Ω

2gdµ liminfnÑ8

»Ω

| fn f |dµ

»

Ω

2gdµ limsupnÑ8

»Ω

| fn f |dµ

Using the fact that³

Ω2gdµ 8, we may cancel the above inequality to obtain

limsupnÑ8³

Ω| fn f |dµ

¤ 0 and so limnÑ8³

Ω| fn f |dµ 0.

Remark 2.1.1. Here are some remarks about the Lebesgue dominated convergence

theorem

(i) The hypothesis can be relaxed becomes fn Ñ f a.e. or | fn| ¤ g a.e.

(ii) By the triangle inequality, it is clear that limnÑ8³

Ωfndµ ³

Ωf dµ .

Theorem 2.1.11 (Beppo-Levi). Let pΩ,F ,µq be a measure space and fn : ΩÑr0,8sbe a sequence of nonnegative measurable functions. Then

»Ω

¸nPN

fndµ ¸nPN

»Ω

fndµ.

19


Proof. Let gN °Nn1 fn and g °

nPN fn. By the monotone convergence theorem,

we have

»Ω

gdµ »

Ω

limNÑ8

gNdµ limNÑ8

»Ω

gNdµ limNÑ8

N

n1

»Ω

fndµ 8

n1

»Ω

fndµ.

B. Tools from Functional Analysis

Definition 2.2.1 (Normed space). A normed space is a vector space X over a field

F P tR,Cu equipped with a function : X Ñ R, called a norm, such that for every

x,y P X and λ P F the following axioms hold

(i) (Nullity) x 0 implies x 0.

(ii) (Homogeinity) λx |λ |x.

(iii) (Triangle inequality) x y ¤ xy.

From these axioms, we can easily deduce that x ¥ 0 and x 0 if and only if x 0.

Definition 2.2.2 (Metric space). A metric space is a nonempty set X equipped with a

function d : X X Ñ R, called a metric, such that for every x,y,z P X the following

axioms hold

(i) (Nullity) dpx,yq 0 if and only if x y.

(ii) (Symmetry) dpx,yq dpy,xq.

(iii) (Triangle inequality) dpx,yq ¤ dpx,zqdpz,yq.

From the third axiom, we also can easily deduce that dpx,yq ¥ 0 for every x,y PX . Also

notice that every normed space is metrizable, i.e., we can generate a metric dpx,yq :x y for every x,y P X in a normed space. This is the standard metric that will be

20


used when we talk about some metric-related objects (e.g. convergence of a sequence)

in a normed space.

Definition 2.2.3 (Sequence in Metric Space, Complete Metric Space). A sequence

pxnqnPN in a metric space X is called converges to a point x P X if for every ε ¡ 0, there

exist N P N such that dpxn,xq ε for every n¥ N. A sequence pxnqnPN in X is called a

Cauchy sequence if for every ε ¡ 0, there exist N PN such that dpxn,xmq ε for every

n,m¥ N. In the case where every Cauchy sequence in X converges to a point in X, we

called X a complete metric space.

Notice that although in the real/complex space every Cauchy sequence is convergent,

a Cauchy sequence in a metric space does not need to be convergent. For example,

consider the sequence xn 1n ,n P N in the metric space of real interval p0,1q with the

usual metric dpx,yq |x y| which is Cauchy but not convergent to an element in

p0,1q.

Definition 2.2.4 (Inner Product). Let X be a vector space over a field F. An inner

product on X is a map x, y : X X Ñ F that satisfies the following axioms for any

x,y,z P X and λ P F.

(i) (Conjugate Symmetry) xx,yy xy,xy.

(ii) (Distributive) xx y,zy xx,zyxy,zy.

(iii) (Homogeneity) xλx,yy λ xx,yy.

(iv) (Positive definiteness) xx,xy ¥ 0 and xx,xy 0 if and only if x 0.

Definition 2.2.5 (Banach and Hilbert space). Banach space is a complete normed

space, i.e., a normed space where is also a complete metric space. In the case that

the norm is induced by an inner product, i.e., x xx,xy 12 for some inner product x, y

defined on the normed space H , we call H a Hilbert space.

21


Definition 2.2.6 (Orthonormal Basis of Hilbert Spaces). Let peiqiPI be a collection of

vectors in H . We call peiqiPI an orthonormal basis of H if xei,e jy 0 for i j and

xh,eiy 0 for every i P I implies that h 0.

Denote by CbpH q the space of all continuous bounded mappings from H into R

endowed by the norm f 0 supxPH | f pxq| for any f PCbpH q.

Proposition 2.2.1. CbpH q is a Banach space.

Proof. Let p fnqnPN be a Cauchy sequence in CbpH q. Since | fnpxq fmpxq| ¤ fn fm0, the sequence p fnpxqqnPN is a Cauchy sequence for any x P H . Since R is com-

plete, its limit is in R and hence the pointwise limit f pxq limnÑ8 fnpxq is an R-

valued function. Now we will show that the limit f is a bounded function as fol-

lows: Let ε ¡ 0 and N be a positive integer such that fn fm0 ε for all n,m ¥ N.

Then, the inequality | f pxq| ¤ | f pxq fNpxq| | fNpxq| ¤ f fN0 fN0 holds for

all x PH . But since for n ¥ N, | fnpxq fNpxq| ε for every x, then | f pxq fNpxq| limnÑ8 | fnpxq fNpxq| ¤ ε so that f fN0 supxPH | f pxq fNpxq| ¤ ε . This

shows that the function f is bounded. Let x0 in H . By the continuity of fN , choose

δ ¡ 0 such that | fNpxq fNpx0q| ε whenever x x0 δ . Then if x x0 δ ,

we have | f pxq f px0q| ¤ | f pxq fNpxq| | fNpxq fNpx0q| | fNpx0q f px0q| ¤ f fN0 | fNpxq fNpx0q| fN f 0 ε ε ε 3ε . This shows the continuity of

f . To finish the proof we need to show fn converges in norm, i.e. f fn0 Ñ 0

as n Ñ 8. This is clear by using the triangle inequality as follows: f fn0 ¤ f fN0 fN fn0 ¤ ε ε 2ε so that f fn0 Ñ 0.

Theorem 2.2.1 (The orthogonal decomposition theorem). Let H be a Hilbert space

and S H a closed subspace of H . Then the orthogonal complement SK defined by

SK tx P X : xx,yy 0 for every y P Su is also a closed subspace of H and H can

be represented as the direct sum of S and SK, i.e., H S`SK.

22


Proof. It is clear that SK is a vector subspace of H . Let xn P SK, n P N, be a sequence

of vectors such that xn Ñ x. Using the continuity of an inner product, we have for any

y P S, 0 limnÑ8 xxn,yy xx,yy which implies that x P SK. To prove the second

statement, we need to show that SXSK t0u and SSK H . Clearly, if x P SXSK,

then xx,xy 0 which implies that x 0. The case that SH is trivial since x x0

for every x PH and SK t0u. Assume that S H and x PH zS, then we have d infyPS x y ¡ 0. Now take a sequence pynqnPN in S such that d limnÑ8 x yn.We will show that pynqnPN is Cauchy as follows

yn ym2 pyn xqpx ymq2

2px yn2x ym2q2x yn ym2

2px yn2x ym2q4y yn ym

22

¤ 2px yn2x ym2q4d2

which is converges to zero as n,m Ñ 8. This implies that there exist an element

y0 P S such that d x y0. Now consider the decomposition x y0px y0q. Let z

be an arbitrary vector in S, then for any c P R, we have y0 cz P S and hence

d2 ¤ px y0q cz2 xpx y0q cz,px y0q czy

x y02 cxx y0,zy cxz,x y0y c2z2

d22cRexx y0,zy c2z2.

Therefore, we have 2cRexx y0,zy c2z2 ¥ 0 and so Rexx y0,zy 0 since c is

arbitrary. On the other side, we also have Imxx y0,zy Rexx y0, izy 0 which

shows that x y0 P SK.

Definition 2.2.7 (Linear Map). Let X and Y be two vector spaces over a field F. A

map T : X Ñ Y is called a linear map or a linear operator if for every x1,x2 P X and

23


c1,c2 P F, we have T pc1x1 c2x2q c1T px1q c2T px2q. We will call a linear map

T : X Ñ F as a linear functional.

Now since a normed space has both topological and algebraic structure, we can

talk about boundedness and continuity of a linear operator as follows.

Definition 2.2.8 (Bounded linear map). Let pX , Xq and pY, Y q be two normed

spaces. A linear map T : X Ñ Y is called bounded if there exist M ¡ 0 such that

T x ¤M for every x P X with x ¤ 1. We denote the smallest such M by T and call

it the operator norm of T . Furthermore, we denote the set of all bounded linear maps

from X to Y by the symbol L pX ,Y q. For X Y , we denote such a set by L pXq.

It is easy to see that this definition is equivalent with saying that T maps every bounded

sets to bounded sets. If the codomain of the map T is complete, i.e., if it is a Banach

space, then the space L pX ,Y q form a Banach space under operator norm as stated in

the following proposition.

Proposition 2.2.2. If X is a normed space and Y is a Banach space, then L pX ,Y q is

a Banach space with respect to the operator norm T supx¤1 T x.

Proof. Let L pX ,Y q be a vector space over field F P tR,Cu such that for any bounded

linear operator T,S we have pαTβSqpxqαT pxqβSpxq for any α,β PF . It is easy

to show that L pX ,Y q together with the operator norm is a normed space. To show

its completeness, let pTnqnPN be a Cauchy sequence in L pX ,Y q. Then for any x P X ,

the sequence pTnxqnPN is a Cauchy sequence in Y because of the following inequality:

TnxTmx ¤ TnTmx and TnTm Ñ 0 as m ¥ n and n Ñ8. It follows that

limnÑ8Tnx exist for any x PX . Denote the limit by T x, i.e., T x limnÑ8Tnx for x PX . To show that T is indeed a member of L pX ,Y q, we first show the linearity of T . Let

x,y P X , then T px yq limnÑ8Tnpx yq limnÑ8pTnxTnyq limnÑ8TnxlimnÑ8Tny T xTy. Next, to show its boundedness, notice that since pTnqnPN is

24


Cauchy in L pX ,Y q, then pTnqnPN is also Cauchy in R by the triangle inequality.

Also, for each x P X we have the inequalities T x ¤ supnPN Tnx ¤ psupnPN Tnqxwhich shows that T is bounded and T ¤ supnPN Tn.

Now we will prove an important proposition that gives a connection between bounded

and continuous operator.

Proposition 2.2.3. Let pX , Xq and pY, Y q be two normed spaces and T : X Ñ Y

a linear map. Then the following statements are equivalent

(i) T is uniformly continuous on X.

(ii) T is continuous at zero.

(iii) T is bounded.

Proof. (i) ñ (ii) trivial.

(ii) ñ (iii) Assume the contrary, i.e., T is continuous at zero but unbounded. The

unboundedness of T implies that for every n P N, there exist xn P X such that xn ¤ 1

but T xn ¡ n. Since xnn ¤ 1

n for every n, then xnn converges to zero in X and hence

our assumption that T is continuous at zero implies that T xnn converges to zero. But

this contradict the fact that T xn ¡ n for every n.

(iii) ñ (i) Assume that T is bounded, then T x ¤ T for every x ¤ 1. Since xx ¤ 1

for every x P X , then we have T xx ¤ T . Consequently, T x ¤ T x and by

additivity of T , T x Ty T px yq ¤ T x y. Therefore, T is a Lipschitz

continuous function which is uniformly continuous.

Definition 2.2.9 (Dual space). Let X be a normed space over field F. The (topological)

dual space of X is the space L pX ,Fq, i.e., the space of all bounded linear functionals

on X and denoted by X.

25


Definition 2.2.10 (Separable space). A metric space X is called separable if it contains

a countable dense subset, i.e., there exist a sequence pxnqnPN in X such that for every

x P X and ε ¡ 0, there exist n P N such that dpxn,xq ε .

Now we will present one of the famous theorem in Hilbert space that we will use

extensively throughout this thesis.

Theorem 2.2.2 (Separability of Hilbert space). A Hilbert space H is separable if and

only if it admits a countable orthonormal basis.

Proof. (ñ) First assume that H is separable, then there exist a sequence pxnqnPN H

dense in H . Now we will construct a maximal subsequence B of pxnqnPN which is

linearly independent by using the induction method. Let n1 be the least positive in-

teger such that xn1 is nonzero. Next, assume that k 1 linearly independent vectors

have already been selected. If for all n ¡ nk1, the vectors xn1 , ...,xnk1,xn are lin-

early dependent, then we stop since we already get a maximal linearly independent

subsequence of pxnqnPN which is finite in this case. On the other side, if there exist

the smallest index nk ¡ nk1 such that the vectors xn1, ...,xnk1,xnk are linearly inde-

pendent, then replace xn1 , ...,xnk1 with the new subsequence xn1, ...,xnk1,xnk . Indeed,

using this construction we finally can obtain a maximal subsequence B of pxnqnPN, i.e.,

B pxn1, ...,xnk1, ...q which is linearly independent. Then, apply the Gram-Schmidt

process to transform the sequence of linearly independent vectors B into a sequence

of orthonormal vectors B0 pun1, ...,unk1, ...q which satisfies spanB spanB0. Also,

since B is a Hamel basis (i.e., a maximal linearly independent subset of pxnqnPN), then

we obtain the following equalities

spanB0 spanB spantpxnqnPNu

but this implies that the linear span of B0 is dense in V and hence its orthogonal com-

plement is only the zero vector. Therefore, B0 is a countable orthonormal basis for H.

26


(ð)Now assume that punqnPN is a countable orthonormal basis for H , then for every

h PH can be represented as

h 8

n1

knun

where the series being finite if H is finite dimensional. Consider the set

B # 8

n1

rnun : rn pn iqn and pn,qn PQ+.

It is a direct consequence of the Cantor diagonalization theorem that the set Q2n tppp1,q1q,pp2,q2q, ...,pp2n,q2nqq : pi,qi PQu is countable for every positive integer n

and hence we can regard B as a countable union of countable sets Q2n in order to

deduce that B is countable. Furthermore, since Q is dense in R, then for any ε ¡ 0,

there exist rn pn iqn where pn,qn PQ such that |kn rn| ¤ ε

2n . Therefore,

h8

n1

rnun ¤8

n1

|kn rn|un ¤8

n1

ε

2n ε,

which shows that the set B is dense in H .

Suppose that pe jq jPN is an orthonormal basis for H . Then we can represent any

vector x P H in the form of the series x °jPN x je j limNÑ8

°Nj1 x je j. By the

orthonormality of the basis vectors pe jq jPN, we have x°Nj1 x je j,eky xk so that xk

xx,eky as N Ñ 8. Therefore, we can obtain the following (generalized) Fourier

series representation of x, i.e., x°jPNxx,e jye j where x j xx,e jy is called the Fourier

coefficients of x.

Proposition 2.2.4 (Parseval Identity). Let pe jq jPN be an orthonormal basis for the

Hilbert space H . Then, for any x PH , we have°

jPN |xx,e jy|2 x2.

27


Proof. Consider the Fourier series representation of the vector x. Then we have

x2 xx,xy x¸jPN

x je j,¸jPN

x je jy limNÑ8

xN

j1

x je j,N

j1

x je jy

limNÑ8

N

j1

x jx j ¸jPN

x jx j

¸jPNxx,e jyxx,e jy

¸jPN

|xx,e jy|2

Definition 2.2.11 (Banach algebra). Banach algebra is a Banach space A which is also

an associative F-algebra, F P tR,Cu, i.e., an additive Abelian group A which has the

structure of both unital ring and F-module such that the scalar multiplication satisfies

αpxyq pαxqy xpαyq for every α P F and x,y P A. Moreover, the norm and the

algebra multiplication satisfies the inequality: xy ¤ xy for every x,y P A.

Definition 2.2.12 (Unital sub-algebra and separating points). Let K be a compact

metric space and CpK,Rq be a Banach algebra equipped with the sup-norm f 0 supxPK | f pxq|. We say that A CpK,Rq a unital sub-algebra if 1 P A and if α f βg, f g P A for any f ,g P A and α,β P R. We say that A separates point of K if for any

two elements x y of K, there exist f P A such that f pxq f pyq.

Definition 2.2.13 (Lattice). Let K be a compact metric space and A a subset of

CpK,Rq. We say that A is a lattice if for every f ,g P A , we have f _ g and f ^ g

are in A , where p f _gqpxq maxt f pxq,gpxqu and p f ^gqpxq mint f pxq,gpxqu.

Before we state the Stone-Weierstrass theorem, we will prove the following useful

lemma about closed sub-algebras.

28


Lemma 2.2.1. Let A CpK,Rq be a closed unital sub-algebra. Then we have the

following three statements

(i) If f is a nonnegative function in A, then?

f is also an element of A.

(ii) If f P A, then | f | P A.

(iii) A is a lattice.

Proof. (i) Since A is closed under scalar multiplication then we can assume that

0 ¤ f ¤ 1 by normalization. Hence, f can be written as f 1 g for some

g P A such that 0 ¤ g ¤ 1. Using a Maclaurin expansion,?

f can be written asaf pxq a

1gpxq 1°8n1 cngnpxq where

cn p1qn11

2n

p1qn1

n!

n1¹j0

p12 jq 212n p2n2q!

n!pn1q!

This Maclaurin series approximates?

f uniformly in 0. Indeed,

a f

18

n1

cngn

0

supxPK

a f pxq

1N

n1

cngnpxq

supyPr0,1s

a1 y

1N

n1

cnyn

.Let ε ¡ 0, by the uniform convergence of the series expansion of

?1 x on

r0,1s, there exist a positive integer N0 such that? f

1°8n1 cngn

0 ε

for any N ¥ N0, i.e., limNÑ8? f

1°N

n1 cngn

0 0. Since for every

nonnegative integer n, we have gn P A, then 1°Nn1 cngn P A since A is a sub-

algebra. Moreover, by the closedness of A, we conclude that?

f P A as desired.

(ii) This result is obtained by substituting f 2 instead of f in (i).

29


(iii) This result comes directly from the following identities and by applying (ii)

f _g 12p f g| f g|q and f ^g 1

2p f g| f g|q.

The following theorem proved by Marshall H. Stone in 1937 generalizes the Weier-

strass approximation theorem which states that the polynomials are uniformly dense in

the space of countinuous real-valued function on ra,bs, Cpra,bs,Rq.

Theorem 2.2.3 (Stone-Weierstrass theorem). Let K be a compact metric space and

A CpK,Rq a unital sub-algebra which separates points of K. Then A is dense in

CpK,Rq.

Proof. Let f PCpK,Rq and ACpK,Rq be a closed unital sub-algebra which separates

points of K. Let ε ¡ 0 be given, we will show that there exists g PA such that f g0 ε . Take arbitrary two different points t1, t2 P K. Since A separates points, there exists

h P A such that hpt1q hpt2q. Now for s1,s2 PR, define h : K ÑR given by hpxq s2ps1 s2q hpxqhpt2q

hpt1qhpt2q for every x P K. Clearly h P A, hpt1q s1, and hpt2q s2. Therefore,

for any t1 t2, there exists ft1,t2 P A such that ft1,t2pt1q f pt1q and ft1,t2pt2q f pt2q.Since ft1,t2 is continuous, ft1,t2 approximates f in neighborhoods of t1 and t2. Now let

t1 be fixed and t2 vary. Define At2 tx P K| ft1,t2pxq f pxq εu, then At2 is open as a

preimage of an open set. Moreover, it also contains t2 which implies that tAt2ut2PK is

an open cover of K. Since K is compact then there exists a finite subcover At2,1, ...,At2,n

such that K nj1 At2. j . Define ht1 min1¤ j¤n ft1,t2, j , then from Lemma 2.2.1, we

have ht1 P A. Clearly, we also have ht1pt1q f pt1q and ht1 f ε . Now define an open

set Bt1 tx P K|ht1pxq ¡ f pxq εu. In a similar way as before, we have K t1PK Bt1 .

Again, by the compactness of K, there exists a finite subcover such that K mj1 Bt1, j .

Put gmax1¤ j¤m ht1, j , then g P A and f ε g f ε , i.e., f g0 ε . Thus, A is

dense in CpK,Rq and hence A CpK,Rq by the closedness of A.

30


Definition 2.2.14 (Baire categories). Let X be a topological space. We say that X is of

the first category if X nPNXn where Xn is a nowhere dense subset (i.e., a set whose

closure has empty interior) of X. Otherwise, we say that X is of the second category.

Lemma 2.2.2. Let X and Y be two normed spaces and T PL pX ,Y q be such that the

range of T , RpT q, is of the second Baire category in Y. If N0X is a neighborhood of the

zero vector 0X in X, then the closure T pN0X q is a neighborhood of the zero vector 0Y

in Y .

Proof. Let 0X ,0Y be the zero vector of X and Y , respectively. It is clear that for an

arbitrary neighborhood N0X of 0X , we can always find a ball B Bεp0Xq such that

BB N0X by taking the radius ε small enough. Also, since for every x P X , we

have limnÑ8 1nx 0, then for every x P X , there is a corresponding positive integer

nx such that x P nxB. This implies that the space X can be written as the union X nxPNpnxBq and hence the range of T , RpT q

nxPNT pnxBq. From the hypothesis,

since RpT q is of the second category, then there exist n0 P N such that T pn0Bq is not

nowhere dense which means that it has nonempty interior. Because of the continuity

of multiplication by a nonzero scalar and multiplication by a nonzero scalar c has

a continuous inverse, namely multiplication by 1c , then multiplication by a non-zero

scalar is a homeomorphism so that T pn0Bq n0T pBq n0T pBq. This shows that the

interior of T pBq is not empty. Therefore, there exist y0 P T pBq such that Bδ py0q T pBqfor some δ ¡ 0. Now consider the ball B Bδ , then we have B y0Bδ py0q y0T pBq T px0qT pBq T px0Bq T pN0X q since x0B BB N0X

and the conclusion follows.

In the following we will state one of the fundamental result in functional analysis

which is also known as the Banach-Schauder theorem.

Theorem 2.2.4 (The open mapping theorem). Let X and Y be two Banach spaces and

T a nontrivial surjective map in L pX ,Y q, then T is an open map, i.e., T pAq is open

31


for every open set A in X.

Proof. We shall denote Xε Bεp0Xq for the open ball in X centered at 0X with radius

ε , and Yε the same in Y . For an arbitrary ε ¡ 0, let εn ε

2n ,n t0uYN. Using Lemma

2.2.2, there exist λn where λn Ñ 0 as n Ñ 8 such that Yλn T pXεnq. Let y P Bλ0 .

Let us claim that there exist an element x P X2ε0 such that T x y. Indeed, from the

above assumption, we know that there exists x0 P Xε0 such that yT x0 ¤ λ1 so that

yT x0 P Yλ1 . Also, there exists x1 P Xε1 such that yT x0T x1 λ2. In a similar

way, we obtain a sequence xn P Xεn such that yT p°ni0 xiq λn1. If yn

°nk1 xk,

then ynym °n

km1 xk ¤°n

km1 xk ¤°n

km1 εk ¤°n

km1 2k

ε0 which

shows that yn is Cauchy. Hence, by the completeness of X ,yn Ñ x as n Ñ8. Also,

since norm is continuous, we have x limnÑ8 °n

k0 xk ¤ limnÑ8°n

k0 xk ¤°nk0 2k

ε0 2ε0. Passing to the limit as n Ñ 8, we get y T x 0 so that

y T x. Since y is an arbitrary element of Bλ0 , then we have shown that Bλ0 T pX2εq.Let G be a nonempty set in X and let x P G. By the openness of G, there exist ε ¡ 0

such that xX2ε G. Consequently T xBλ0 T xT pX2εq T pxX2εq T pGq,i.e., for any x P G, T pGq contains a neighborhood of T x which shows that T pGq is

open.

The following theorem is a direct consequence of the open mapping theorem.

Theorem 2.2.5 (Bounded inverse theorem). Let X ,Y be two Banach spaces and T PLpX ,Y q is bijective, then T is a homeomorphism.

Proof. Since RpT q Y , then T is open by the open mapping theorem, so that the

preimage of every open set in X under T1 : Y Ñ X is an open set in Y and the conclu-

sion follows.

If X ,Y are two normed spaces with norms X and Y , respectively, then X Y is also a normed space with respect to the norm (not unique) px,yq pxp

X

32


ypY q

1p for 1 ¤ p 8. Until now we only consider linear transformations from a

vector space X to a vector space Y , i.e., their domain of definition coincide with the

entire space X . Now we will also consider a transformation defined on a proper sub-

space of X which we also called an operator.

Definition 2.2.15 (Graph, closed, and closable operators). Let X and Y be two normed

spaces and T : DpT q X Ñ Y an operator defined on its domain of definition DpT q.The graph of T , denoted by GpT q, is defined as GpT q graph T tpx,T xq : x P DpT qu X Y . Moreover, the operator T is said to be closed if its graph GpT q is a closed

subspace of X Y . If the closure of graph of T , GpT q X Y , can be identified as a

graph of a linear operator T , then T is said to be closable and we call T the closure of

T .

Proposition 2.2.5. Let X ,Y be two normed spaces and T : DpT q X Ñ Y a linear

operator. Then T is closable if and only if for every sequence pxnqnPN DpT q such

that xn Ñ 0 and T xn Ñ y implies y 0.

Proof. Let pxnqnPN be a sequence in DpT q such that xn Ñ 0 and T xn Ñ y, then pxn,T xnqÑp0,yq as n Ñ8. Let T be the closure of T , then we have p0,yq P GpT q GpT q so

that y T p0q 0 by the linearity of T . For the converse, define

DpUq !

x P X : Dy P Y,px,yq P GpT q),

i.e., the projection of GpT q on X . It is easy to see that DpUq is a subspace of X . Let

x P DpUq and assume that there exist y1,y2 P Y such that px,y1q,px,y2q P GpT q. Then,

there exist two sequences pn,qn in DpT q such that ppn,T pnqÑ px,y1q and pqn,T qnqÑpx,y2q as n Ñ8. By letting rn pn qn, we have rn Ñ 0 and Trn Ñ y1 y2 0

by our hypothesis. Therefore, y1 y2 which shows that for every x P DpUq there

exist a unique y such that px,yq P GpT q. Now define a linear operator U : DpUq Ñ Y

33


where Sx y such that px,yq P GpT q. Similar as before, let p,q P DpUq, then there

exist two sequences pn,qn in DpT q such that pα1 pnα2qn,T pα1 pnα2qnqqÑ pα1 pα2q,α1U pα2Uqq. This shows the linearity of U , and the conclusion follows.

We will state a fundamental result concerning closed operators due to Banach

which shows that the domain of every noncontinuous closed operators T 0 cannot

be the entire space.

Theorem 2.2.6 (Closed graph theorem). Let X and Y be Banach spaces and T a closed

linear operator from X to Y with DpT q X, then T is continuous.

Proof. If pxn,ynqnPN is a Cauchy sequence in XY , then pxnqnPN is a Cauchy sequence

in X and pynqnPN is a Cauchy sequence in Y . Hence if both X and Y are complete, then

there exist x P X and y P Y such that pxn,ynq Ñ px,yq by using the definition of norm

in the Cartesian space X Y . Thus, we conclude that if X and Y are Banach spaces,

then X Y is also a Banach space. Moreover, since the operator T is closed, then

GpT q is a closed subspace of a complete space X Y which is also complete. Define a

projection pX : GpT qÑX where pXppx,T xqq x, then clearly pX is a bijection between

GpT q and X and is continuous. Moreover, the bounded inverse theorem implies that

pX has a continuous inverse p1X . Define also the second projection pY : GpT q Ñ Y

defined by pY ppx,T xqq T x. Then, clearly we have T pY p1X which is continuous

as a composition of two continuous operators.

From now on, H will denote a separable Hilbert space and will be the norm

induced by the inner product x, y on H . Furthermore, BpH q denote the Borel σ -

algebra on H . Let LpH q be the set of all bounded linear operator on H .

Definition 2.2.16 (Trace class operator). A trace class operator is an operator T PLpH q such that there exist two sequences pa jq jPN,pb jq jPN in H with

°8j1 |a j||b j|

8 and T x °8j1 a jxx,b jy. If pe jq jPN is an orthonormal basis for H , then for any

34


linear operator T on H we define the trace of T by

Tr T 8

j1

xTe j,e jy.

Proposition 2.2.6. If T P LpH q is a trace class operator, then

(i) T is compact.

(ii) For any orthonormal basis pe jq jPN in H , Tr T=°8

j1xTe j,e jy is absolutely con-

vergent and equal to°8

j1xa j,b jy.

Proof. First, recall that an operator T on H is compact if and only if T is a strong limit

of a sequence of finite rank operators pTnqnPN in H . Choose Tnx °nj1 a jxx,b jy.

Then by using both triangle and Cauchy-Schwarz inequalities, it is easy to see that

T Tn ¤8

jn1

|a j||b j|,

where is the operator norm on H . From the definition of a trace class operator, it

is obvious that right hand side of the inequality converges to zero as n Ñ8.

For the second statement, again it is easy to show that after some computations

using the additivity and homogeneity property of an inner product and then using the

Cauchy-Schwarz inequality we obtain

8

j1

|xTe j,e jy| ¤8

j1

8

k1

|xa j,eky|212 8

k1

|xb j,eky|212

.

From this point, by applying the Parseval identity yields

8

j1

8

k1

|xa j,eky|212 8

k1

|xb j,eky|212

8

j1

|a j||b j| 8,

i.e. the sum is absolutely convergent. The last part that°8

j1xTe j,e jy °8

j1xa j,b jy

35


can be proven by rewritting°8

j1xTe j,e jy as

8

k1

x8

j1

xak,e jye j,bky

and then using the fact that pe jq jPN is an orthonormal basis in H .

We will denote the set of all positive, symmetric, trace class operator Q on H by the

symbol L1 pH q.

Definition 2.2.17 (Fréchet derivative). Let X ,Y be two normed spaces. The Fréchet

derivative (or strong derivative) of a function f : X Ñ Y at x is a bounded linear map

D f pxq : X Ñ Y such that

limhÑ0

f pxhq f pxqD f pxqhh 0.

We will denote the k-th order derivative of f by the symbol Dk f .

This definition can be seen as generalization of the derivative of a function f : RÑ R

and the Jacobian of a function f : Rn Ñ Rm to the derivative in an arbitrary normed

space.

Let us now recall the Riesz representation theorem which is very important in the

construction of a measure in an infinite dimensional spaces.

Theorem 2.2.7 (The Riesz representation theorem). Let H denote the dual space of

a separable Hilbert space H consisting of all continuous linear functionals from H

into its field. Then for any element T of H , there exist a unique element y PH such

that

T pxq xx,yy, @x PH

Furthermore, T y.

36


Proof. Let T PH . First we will show that the representation of T is well-defined. Let

y1,y2 PH be two elements such that T pxq xx,y1y xx,y2y, then we have xx,y1 y2y0. Setting x y1 y2, we obtain y1 y22 0 which implies that y1 y2. Since

H is separable, there exists an orthonormal basis pe jq jPN for H . Let T P H and

a j T pe jq for j 1, ...,n, .... Let x P H and c j @

x,e jD

. Define xn °n

j1 c je j,

then clearly we have limnÑ8 x xn 0. From the linearity of T , we have T pxnq °nj1 a jc j. Consider the inequality T pxqT pxnq ¤ T x xn. Since T is bounded

and limnÑ8 x xn 0, then we have T pxq limnÑ8T pxnq °8

j1 a jc j. In fact,

the sequence a j must itself be square summable. To see this, first note that since

T pxq ¤ T x, then we have |°8j1 a jc j| ¤ T

°8j1 c2

j

12 for any sequence c j.

Now let N be a fixed positive integer and define a sequence

c j

$''&''%a j if j ¤ N

0 if j ¡ N,

then the last inequality yields |°Nj1 a2

j | ¤ T °N

j1 a2j

12 which is equivalent to°N

j1 a2j

12 ¤ T . Clearly, this shows that the sequence pa jq jPN is square-summable

by the least upper bound property of R. This implies that, as an element of H ,

the series y °8j1 a je j is well-defined which means that we can have the equality

T pxq °8j1 a jc j xx,yy. Moreover, by taking the limit as n Ñ8 on the inequal-

ity°N

j1 a2j

12 ¤ T , we have y ¤ T . On the other side, the Cauchy-Schwarz

inequality implies that |T pxq| |xx,yy | ¤ xy which is equivalent to |T pxq|x ¤ y.

Thus, T y.

Theorem 2.2.8 (Riesz Representation Theorem for Bilinear Form). Let ϕ be a bounded

bilinear form on H . Then, there exist a unique bounded linear operator Q P LpH qsuch that ϕpx,yq xQx,yy for every x,y PH .

37


Proof. Obviously, if Q exist, then it is unique. Now let x PH be fixed. Then, ϕpx,yqis conjugate linear in y, so that ϕpx,yq is linear in y. Using the Riesz representation

theorem, there exist a unique z P H such that ϕpx,yq xy,zy, and hence ϕpx,yq xz,yy. We now define the map Q : H ÑH by Qx z. Since

xQpαx1βx2q,yy ϕpαx1βx2,yq αϕpx1,yqβϕpx2,yq

αxQx1,yyβ xQx2,yy

xαQx1,yyxβQx2,yy

This implies that xQpαx1βx2q,yyxαQx1,yyxαQx2,yy PH K t0u so that Q is

linear. To show that Q is bounded, notice that

Q supx0

Qxx sup

x0,Qx0

xQx,QxyxQx

¤ supx0,y0

xQx,yyxy

supx0,y0

ϕpx,yqxy

ϕ,


We are going to use both of the Riesz representation theorem to define the mean

and covariance of a probability measure µ in the next chapter.

C. Events and Random Variables

We say that two events A and B are (stochastically) independent if the occurence of A

does not change the probability that the event B also occurs. In a more precise way, we

have the following

38


Definition 2.3.1 (Independence of events). Let I be an arbitrary index set. A collection

of events pAiqiPI is called independent if for every finite subset J I, we have

P

£jPJ

A j

¹jPJ

PpA jq

Now suppose that we roll a dice infinitely many times, what is the probability that

the face shows a one infinitely many ? We guess that this should be equal to one since

otherwise there would be a last point in time when we see a one and after which the face

only shows a number two to six. However, this is not very plausible. The following

theorem confirms the conjecture we mentioned before and also gives conditions under

which we cannot expect that infinitely many of the events occur.

Theorem 2.3.1 (Borel-Cantelli lemma). Let pAnqnPN be a sequence of events and A limsupnÑ8An. Then

(i) Let µ be an arbitrary measure. If°8

n1 µpAnq 8, then µpAq 0.

(ii) If pAnqnPN is independent and°8

n1PpAnq 8, then PpAq 1.

Proof. By the continuity from above and σ -subadditivity of µ , respectively, we have

µpAq limnÑ8µ

8¤mn

Am

¤ lim

nÑ8

8

mnµpAmq 0.

For the second part, by using the extended De’Morgan’s rule and continuity from below

of P, we have

PpAcq P

8¤m1

8£nm

Acn

lim

mÑ8P

8£nm

Acn

However, for every m PN, P8

nm Acn±8

nmp1PpAnqq since the independence of

pAnqnPN implies the independence of pAcnqnPN. Also, since lnp1xq ¤ x for x P r0,1q,

39


then we have the following inequality

8¹nm

p1PpAnqq expp8

nmlnp1PpAnqq ¤ expp

8

nmPpAnqq 0


Now we extend the definition of independence from a collection of events to a

family of collection of events.

Definition 2.3.2 (Independence of a collection of events). Let pΩ,F ,Pq be a proba-

bility space and I be an arbitrary index set. Let Ai F for every i P I. The collection

pAiqiPI is called independent if for any finite subset J of I and any choice A j PA j, we

have

P

£jPJ

A j

¹jPJ

PpA jq.

Let pΩ,F ,Pq be a probability space, pΩ,F q a measurable space, and BpΩq the

Borel σ -algebra of Ω.

Definition 2.3.3 (Random variable). Let Ω be a complete metric space. An Ω-valued

random variable defined on the measure space pΩ,F ,Pq is a measurable function

X : Ω Ñ Ω, i.e, X1pIq P F for every I P BpΩq. If Ω R, then we call X as a real

random variable.

Remark 2.3.1. For F PBpΩq, we denote tX P FuX1pFq and PpX PFqPrX1pFqs.In particular, we let tX ¥ au X1pra,8qq, tX au X1pp8,aqq, and so on.

Definition 2.3.4 (Distribution of random variables). Let X be a random variable. Then

(i) The probability measure PX PX1 is called the distribution of X.

(ii) For a real random variable X, the function FX which maps x into PpX ¤ xq is

called the distribution function of X. We write X µ if X has distribution µ , i.e,

µ PX .

40


(iii) Let I be an arbitrary index set. A collection pXiqiPI is called identically dis-

tributed if PXi PX j for every i, j P I.

Using the concept of independence of a collection of events, we can also define

independence of random variables via independence of the σ -algebras they generate.

Definition 2.3.5 (Independence of random variables). The collection pXiqiPI of random

variables is called independent if the collection pσpXiqqiPI is independent.

Remark 2.3.2. We call a collection of random variables pXiqiPI is i.i.d. if pXiqiPI is

independent and identically distributed.

Remark 2.3.3. It is clear that pXiqiPI is independent if and only if

P

£jPJ

A j

¹jPJ

PpA jq

for any finite set J I and any choice A j P F j, j P J.

D. Motivation to Measures in Infinite Dimensional Spaces

Here we will prove by using the Riesz lemma that we cannot define a Lebesgue mea-

sure, i.e., a locally finite and translation invariant Borel measure, on an infinite dimen-

sional separable Banach spaces.

Lemma 2.4.1 (Riesz’s lemma). Let X be a normed space and Y be a closed proper

subspace of X. If r P p0,1q, then there exist xr P X such that xr 1 and r dpxr,Y q ¤1.

Proof. Let x P XzY , since Y is closed, then dpx,Y q infyPY xy ¡ 0. Since r P p0,1q,we have dpx,Y q 1

r dpx,Y q. Consequently, there exists y0 P Y such that 0 x y0 1r dpx,Y q. Let xr xy0

xy0 , then xr 1 and for every y P Y

infyPY

xr y infyPY

x y0

x y0 y 1x y0 inf

yPYxpy0x y0yq ¡ r.

41


Finally, since 0 P Y , it follows that dpxr,Y q ¤ x 1.

It is easy to show that Riesz’s lemma is equivalent with saying that for any non-dense

subspace Y of a normed space X , given r P p0,1q, then there exist xr P X with xr 1

and r dpxr,Y q ¤ 1. In this context, it gives a condition for a subspace to be dense.

This simple yet very useful theorem can be seen as a substitute for orthogonality when

we are not dealing with a Hilbert space. In a Hilbert space the equality in the Riesz

lemma can be achieved, i.e., dpxr,Y q 1, by taking any unit vector xr in the orthogonal

complement of Y as follows

dpxr,Y q infyPY

xr y infyPYxxr y,xr yy xr22Rexx,yy inf

yPYy2 1.

Now we will show that Lebesgue measure is uniquely determined by the property of

being finite on compacta and invariant under translation. The proof for n-dimensional

case is essentially the same as given below.

Lemma 2.4.2. Let µB be a translation invariant nonnegative Borel measure (i.e., a

measure defined on Borel sets) such that µBpr0,1qq 1 and µ a Lebesgue measure on

R. Then µB cµ for some real constant c.

Proof. First, we denote µpr0,1qq c and observe that the semi-closed interval r0,1qcan be written in the following form

r0,1q

0,1n

Y

1n,2n

Y ....Y

n1

n,1

and hence µBp0, 1

n

q 1n µBpr0,1qq 1

nc by the translation invariance and disjoint

additivity of µB. It is clear that for any rational number r, µBpr0,rqq rc by the similar

idea as above. Now let b be a fixed real number. Choose an increasing sequence of

rational numbers pqnqnPN such that qn Ñ b as n Ñ8. Using continuity from below,

42


we obtain

µBpr0,bqq µBp8¤

n1

r0,rnqq limnÑ8µBpr0,rnqq lim

nÑ8rnc bc.

This implies that µBpra,bqq pb aqc by the translation invariance of µB. Now we

will use the Dynkin π λ theorem to extend this result to an arbitrary Borel set A of

R. Let M be the collection of all intervals ra,bq and N the collection of all Borel sets

of R for which µBpAq cµpAq holds. Clearly M is a π-system. Since we want N to

be a λ -system, then for any A P N we should have Ac P N, i.e., µBpAcq µBpRq µBpAq 8 cµpAq cµpAcq which is nonsense. To get rid of this problem, we

let n be a fixed positive integer and Nn be the collection of all Borel sets A for which

µBpAX rn,nqq cµpAX rn,nqq. From the previous result, we have M Nn. It

is also clear that Nn contains the empty set and is closed under countable disjoint

unions. Moreover, for any A P Nn, we have µBpAcXrn,nqq µBprn,nqqµBpAq cµprn,nqqcµpAq cµpAcXrn,nqq. This shows that Ac P Nn and hence Nn is a λ -

system. Therefore, Dynkin’s πλ -theorem implies that Nn σpMq BpRq, i.e., Nn

is the entire Borel σ -algebra of R. By letting nÑ8, we have

nPNpAXrn,nqq A


One of the most important property of the Lebesgue measure is that it is invariant

under translation. This leads to nice interaction between differentiation and integra-

tion, such as integration by parts. It also gives nice functional-analytic properties to

differential operators. For example, the Laplacian is a self-adjoint operator on the

Hilbert space L2pRnq.

Theorem 2.4.1 (Nonexistence of nontrivial infinite dimensional Lebesgue measure).

The only Lebesgue measure, i.e., translation invariant Borel measure which assign 1 to

the unit ball, on an infinite dimensional separable Banach space is the trivial measure.

Proof. Let BpXq be the Borel σ -algebra on an infinite dimensional Banach space X

43


and µ be a translation invariant Borel measure on the measurable space pX ,BpXqqsuch that µpB1p0qq 1. Since µpB1p0qq 1, then for any x P X there exist δx ¡ 0 such

that µpBδxpxqq 8 by the translation invariance of µ , i.e., µ is locally finite on X .

Hence, there exist δ ¡ 0 such that µpBδ p0qq 8. Now we will construct a countable

collection of pairwise disjoint open balls inside Bδ p0q in the following way:

(i) Take an arbitrary unit vector y1 P X and define Y1 spanty1u. Then there exists

a unit vector y2 P X such that dpy2,Y1q ¡ 12 by the Riesz lemma.

(ii) Assume that we have already ty1, ...,ynu. Let Yn spanty1, ...,ynu. Again by

Riesz’lemma, there exists a unit vector yn1 P X such that dpyn1,Ynq ¡ 12 .

But since the space X is infinite dimensional, we can repeat this construction countably

many to obtain a sequence of unit vectors pynqnPN in X such that yn ym ¡ 12 for all

n m. Now we construct a new sequence pxnqnPN from pynqnPN where xn δ

2 yn,

xn δ

2 , and xn xm ¡ δ

4 for every n m. From this new sequence, construct a

collection of open balls B δ

8pxnq for all n PN. Notice that for every x P B δ

8pxnq, we have

the inequality x ¤ xxnxn 5δ

8 which implies that every ball in the collection

B δ

8pxnq is contained in Bδ p0q. Furthermore, if x P B δ

8pxnqXB δ

8pxmq for n m, then

δ

4 xn xm ¤ xn x x xm δ

4 , which is a contradiction. Therefore, the

collection B δ

8pxnq is pairwise disjoint. Now using the locally finiteness of µ together

with some basic properties of µ , we have

8¡ µpBδ p0qq µp8¤

n1

B δ

8pxnqq

8

n1

µpB δ

8pxnqq

8

n1

µpB δ

8pxmqq

for an arbitrary m P N. The last equality is clear by the translation invariance of

µ . But the inequality°8

n1 µpB δ

8pxmqq 8 forces that µpB δ

8pxmqq 0 and hence

µpB δ

8pxqq 0 by the translation invariance of µ . Since X is separable, there ex-

ist a dense sequence penqnPN of subsets of X . Then for any Borel set A, we have

44


A 8n1 B δ

8penq. Applying our result before together with the monotonicity and σ -

subadditivity of µ , we have

0 ¤ µpAq ¤ µp8¤

n1

B δ

8penqq ¤

8

n1

µpB δ

8penqq 0

which implies that µ 0 on BpXq.

A more intuitive idea why infinite dimensional Lebesgue measure cannot exist

comes from considering the effect of scaling. In Rn, the measure of the ball Brp0qis rn times larger than the ball B1p0q, When n 8 this suggests that one cannot get

sensible numbers for the measures of balls. But we still can have a nontrivial transla-

tion invariant Borel measure on infinite dimensional spaces. For instance, consider the

counting measures which measure the cardinality of the given set. But these measures

are less useful for analysis since they cannot say anything helpful about open sets. This

gives us a reason to construct an infinite dimensional measure which is similar to the

very nice Lebesgue measure without losing too many properties in order to do calculus

in infinite dimensional spaces. We will construct such a measure called the Gaussian

measure in the next chapter.

45


CHAPTER 3

GAUSSIAN MEASURES IN HILBERT SPACES

A. Mean and Covariance of Probability Measures in Hilbert Spaces

Let us now fix a probability measure µ on pH ,BpH qq such that

»H|x|µpdxq 8.

For every h PH we define a linear functional F : H Ñ R as

Fphq »

Hxx,hyµpdxq.

Obviously F is bounded since, by using the Cauchy-Schwarz inequality, for |h| ¤ 1 we

have

|Fphq| »

Hxx,hyµpdxq ¤ |h|

»H|x|µpdxq 8,

i.e. F is a continuous linear functional on H . By Riesz’ representation theorem there

is exactly one m PH such that

xm,hy »

Hxx,hyµpdxq, h PH .

The vector m is called the mean of µ . We shall write

m »

Hxµpdxq.

Now set »H|x|2µpdxq 8,

46


and define a bilinear form G : H H Ñ R as

Gph,kq »

Hxh,xmyxk,xmyµpdxq.

Again, the Cauchy-Schwarz inequality for |h| ¤ 1 and |k| ¤ 1 gives

|Gph,kq| ¤ |h||k|»

H|xm|2µpdxq 8,

by using the fact that³H |x|2µpdxq 8 implies

³H |x|µpdxq 8. Therefore, by

Riesz representation theorem for bilinear form, there exists Q P LpH q such that

xQh,ky »

Hxh,xmyxk,xmyµpdxq, h,k PH .

We called such a Q as the covariance operator of µ .

Proposition 3.1.1. Let µ be a probability measure on pH ,BpH qq with mean m and

covariance Q. Then Q P L1 pH q.

Proof. The positivity and symmetric properties of Q are clear. To prove that Q is of

trace class, we will show that the trace of Q is finite, i.e.

Tr Q 8

j1

xQe j,e jy 8

j1

»H|xe j,xmy|2µpdxq 8.

By using the monotone convergence theorem and the Parseval identity, we obtain

Tr Q »

H

8

j1

|xe j,xmy|2µpdxq »

H|xm|2µpdxq 8.

47


We define the Fourier transform of the measure µ as

µphq »

Heixx,hy

µpdxq, h PH .

The Fourier transform has a nice properties that it characterizes the measure.

B. Law of a Random Variable

Definition 3.2.1 (Law of a random variable). Let Ψ be a complete metric space. Let

X be a Ψ-valued random variable in the measure space pΩ,F ,Pq. The law of a

random variable X is the probability measure X#P on pΨ,BpΨqq defined by X#PpIq PpX1pIqq PpX P Iq for every Borel set I P BpΨq.

Theorem 3.2.1 (Change of variables formula). Let X be an Ψ-valued random variable

in pΩ,F ,Pq and ϕ : Ψ Ñ R be a bounded Borel mapping. Then we have

»Ω

ϕpXpωqqPpdωq »

Ψ

ϕpxqX#Ppdxq

Proof. In the case that ϕ 1I where I P BpΨq, we have ϕpXpωqq 1X1pIqpωq for any

ω P Ω and hence we have the following equalities

»Ω

ϕpXpωqqPpdωq PpX1pIqq X#PpIq »

Ψ

ϕpxqX#Ppdxq

Since ϕ can be approximated by simple functions, then this property also hold for ϕ

by the linearity of Lebesgue integration and the monotone convergence theorem.

Proposition 3.2.1. Let µ1 and µ2 be two probability measures on pH ,BpH qq such

that for any bounded continuous function φ : H Ñ R, we have

»H

φpxqµ1pdxq »

Hφpxqµ2pdxq,

48


then µ1 µ2.

Proof. Let A H be a closed subset of H and pφnqnPN : H Ñ R be a sequence

defined by

φnpxq

$''''''&''''''%1 if x P A

1ndpx,Aq if x ¤ 1n

0 if x ¥ 1n

Since dpx,Aq is continuous on H , then φn is continuous for every n. Obviously, φn is

bounded with supxPH |φnpxq| ¤ 1. Letting n Ñ8 on the equality³H φpxqµ1pdxq ³

H φpxqµ2pdxq, we obtain µ1pAq µ2pAq by the Lebesgue dominated convergence

theorem. Finally, since BpH q is generated by closed sets on H and A is arbitrary,

then the conclusion follows.


that pPnq#µ1 pPnq#µ2 for every n P N. Then µ1 µ2.

Proof. Let φ : H ÑR be a bounded and continuous map. Since φ is continuous, then

by the Lebesgue dominated convergence theorem, we have

»H

φpxqµ1pdxq limnÑ8

»H

φpPnxqµ1pdxq.

Using the change of variables formula, we obtain the following

limnÑ8

»H

φpPnxqµ1pdxq limnÑ8

»PnpH q

φpyqpPnq#µ1pdyq

limnÑ8

»PnpH q

φpyqpPnq#µ2pdyq

limnÑ8

»H

φpPnxqµ2pdxq

»

Hφpxqµ2pdxq.

49


Since the bounded continuous map φ is arbitrary, we obtain the desired result.


that µ1 µ2. Then µ1 µ2.

Proof. For any n P N and h PH , we have

µ1pPnhq »

Heixx,Pnhy

µ1pdxq »

PnpH qeixPny,PnhypPnq#µ1pdyq.

Similarly,

µ2pPnhq »

Heixx,Pnhy

µ2pdxq »

PnpH qeixPny,PnhypPnq#µ2pdyq.

Since µ1 µ2, then pPnq#µ1 and pPnq#µ2 have the same Fourier transform so that

pPnq#µ1 pPnq#µ2. Thus we have µ1 µ2 by the previous proposition.

C. Gaussian Measures

Definition 3.3.1 (Gaussian Measures). Let m PH and Q P L1 pH q, a Gaussian mea-

sure µ Nm,Q on a measurable space pH ,BpH qq is a probability measure µ having

mean m and covariance operator Q, and Fourier transform given by

Nm,Qphq eixm,hy 12 xQh,hy, h PH . (3.1)

If m 0, we denote µ by µ NQ. The Gaussian measure Nm,Q is called non-degenerate

if its kernel, Ker Q th PH : Qh 0u t0u.

Now we are going to show that for any m P H and Q P L1 pH q, there exists a

unique Gaussian measure µ Nm,Q in pH ,BpH qq starting from the finite dimen-

sional case.

50


1. Gaussian Measures in R

Here we shall show that for any m P R and Q ¥ 0, there exists a Gaussian measure

Nm,Q on pR,BpRqq as follows.

Proposition 3.3.1. Let m P R.

(i) Nm,0 δm, where δm is the Dirac measure at m, i.e., a measure defined on BpRqsuch that δmpAq 1Apmq where 1A is the indicator function of A, m is its mean,

and 0 is its covariance.

(ii) Let δ ¡ 0 and µ be defined by µpdxq 1?2πδ

epxmq2

2δ dx ,then µ Nm,δ where

m is its mean, and δ is its covariance.

Proof. (i) It is clear from the definition of Dirac measure that δmpRq 1 since m PRso δm is a probability measure and its Fourier transform is given by

»R

eixhδmpdxq eimh,

It is also clear that »R

x δmpdxq m,

and »Rpxmq2 δmpdxq 0,

which shows that m and 0 are the mean and the covariance of µ , respectively.

(ii) Using the normal transformation z xm?δ

, and the well-known fact that³88 e

z22 dz

?2π , we obtain

1?2πδ

» 8

8e

pxmq2

2δ dx 1?2π

» 8

8e

z22 dz 1,

51


i.e. µ is a probability measure on pR,BpRqq. It also can be checked that

»R

eixhµpdxq eimh 1

2 δh2,

which satisfies the condition p1q for any m P R and Q ¡ 0. Furthermore, it can

be shown by some elementary integration techniques that

»R

xµpdxq m,

which shows that m is the mean of µ and

»Rpxmq2µpdxq δ ,

which shows that Q is the covariance of µ .

2. Gaussian Measures in Rn

Now we recall the notion of product measure. For simplicity we only consider 2-

dimensional product measures, any other finite or infinite product measures are ex-

tendable in a similar manner. Let pH ,BH q and pG,BG q be measurable spaces, then

BH BG is the σ -algebra generated by the sets EF with E PBH and F PBG . In

other words, BH BG is the coarsest σ -algebra with the property that the product of

a BH -measurable set and a BG -measurable set is always BH BG measurable.

Let pH ,BH ,µ1q and pG ,BG ,µ2q be σ -finite measure spaces (i.e. H and G

can be expressed as the countable union of measurable sets of finite measure), then

we define the product measure µ1 µ2 as a measure on the measurable space pH

52


G ,BH BG q such that

pµ1µ2qpEFq µ1pEqµ2pFq

for any E PBH and F PBG .

Proposition 3.3.2. (Existence and uniqueness of product measure)

Let pH,BH ,µ1q and pG,BG ,µ1q be σ -finite measure spaces. Then there exists a

unique measure µ1µ2 on BH BG that satisfies pµ1µ2qpEFq µ1pEqµ2pFqwhenever E PBH and F PBG .

For complete details and proof, see Tao, T. Now we shall show the existence of a Gaus-

sian measure in a finite dimensional Hilbert space H Rn. First recall a well known

fact from linear algebra that since the positive linear operator Q in Rn is symmetric,

then there exists an orthonormal basis pe jq jPNn for Rn consisting of eigenvectors of Q

where Nn 1, ...,n. This implies that there exists nonnegative numbers pδ jq jPNn such

that

Qe j δ je j, j P Nn.

Set

m j xm,e jy, j P Nn,

then we have the following proposition.

Proposition 3.3.3. Let µ be defined as product measure of measures in R, Nm j,δ j , as

follows

µ n¡

j1

Nm j,δ j ,

then µ Nm,Q, m is a Gaussian measure in Rn where m is the mean, and Q is the

covariance of µ .

Proof. Using Fubini’s theorem, it is obvious since each of the Nm j,δ j is a probability

53


measure, then so does µ . Moreover, we obtain

»Rn

eipx1h1...xnhnqµpdxq »Rn

eipx1h1...xnhnqNm1,δ1pdxnq...Nmn,δnpdxnq

n¹

j1

eim jh j 12 δ jh2

j

eixm,hy 12 xQh,hy

Moreover,

»Rn

xµpdxq m and»Rnxy,xmyxz,xmyµpdxq xQy,zy, y,z PH

which shows that m and Q are the mean and the covariance of µ , respectively.

3. Gaussian Measures in Hilbert Spaces

Finally, we are going to define a Gaussian measure µ Nm,Q in an infinite dimensional

separable Hilbert space H for any m P H and Q P L1 pH q. From the spectral theo-

rem, we know that since Q is a compact and symmetric operator on separable Hilbert

space H then there exists an orthonormal basis pe jq jPN of H consisting of eigen-

vectors of Q. Recall that Q is positive which implies that there exists a nonnegative

sequence pδ jq jPN such that

Qe j δ je j, j P N.

Now for any x P H we set

x j xx,e jy j P N.

As a consequence of the Parseval identity, it is clear that since pe jq jPN is an orthonormal

basis of H , then the map Ψ : H Ñ `2 defined by

Ψpxq pxx,e jyq jPN, x PH

54


is well-defined. Moreover, it is an isomorphism between H and the space of all square

summable sequences `2. Furthermore, since H is separable, then H is isometrically

isomorphic to `2 space. This means that, since H and `2 is isometrically isomorphic,

we can identify H with `2.

Let µ be defined as a countable product of measures in R as follows

µ 8¡

j1

Nm j,δ j .

Now, we shall show that though µ is defined on the space R8 instead on `2, but µ is

concentrated on `2.

Proposition 3.3.4. We have µp`2q 1.

Proof. Using the monotone convergence theorem, we have

»R8

8

j1

x2j µpdxq

8

j1

»R

x2jNm j,δ jpdx jq.

Now notice that since

δ »Rpxmq2µpdxq

»R

x2µpdxq2m

»R

xµpdxqm2

»R

x2µpdxq2m2m2

then, »R

x2µpdxq δ m2.

Therefore,

8

j1

»R

x2jNm j,δ jpdx jq

8

j1

δ jm2j Tr Q|m|2 (3.2)

55


and thus

µpx P R8 : |x|2`2 8q 1

as desired.

Theorem 3.3.1. µ is a Gaussian measure with mean m and covariance Q, i.e. µ Nm,Q where m pm jq jPN and Qx pδ jx jq jPN.

Proof. First for the orthonormal basis pe jq jPN we set the orthogonal projection operator

from H to PnpH q, i.e.

Pnx n

j1

xx,e jye j, x PH .

Since pe jq jPN is an orthonormal basis of the separable Hilbert space H , then for any

x PH , the equality

limnÑ8Pnx x

holds. Now using the Lebesgue dominated convergence theorem we obtain

»H

eixx,hyµpdxq lim

nÑ8

»H

eixPnx,hyµpdxq

limnÑ8

n¹j1

»R

eix jh jNm j,δ jpdx jq

limnÑ8

n¹j1

eim jh j 12 δ jh2

j

limnÑ8eixPnm,hye

12 xPnQh,hy

eixm,hy 12 xQh,hy

i.e. µ Nm,Q as desired.

Furthermore, from (3.2), it follows that

»H|x|2µpdxq Tr Q|m|2 8

56


i.e. the second moment of µ is finite. Therefore, we have by the Lebesgue dominated

convergence theorem

»Hxx,hyµpdxq lim

nÑ8

»HxPnx,hyµpdxq

limnÑ8

n

k1

»H

xkhkµpdxq

limnÑ8

n

k1

hk

»R

xkNmk,δkpdxkq

limnÑ8

n

k1

hkmk

limnÑ8xPnm,hy

xm,hy,

which shows that m is the mean of µ , and

»Hxy,xmyxz,xmyµpdxq lim

nÑ8

»Hxy,Pnpxmqyxz,Pnpxmqyµpdxq

limnÑ8

n

k1

»H

ykzkpxkmkqµpdxq

limnÑ8

n

k1

ykzk

»R

xkmkNmk,δkpdxkq

limnÑ8

n

k1

ykzkδk

limnÑ8xPnQy,zy

xQy,zy,

which shows that Q is the covariance of µ .

Now we will present some computations of Gaussian integral involving a measure

in an infinite dimensional separable Hilbert space H .

Proposition 3.3.5. Let µ Nδ be a Gaussian measure in H , then there is an or-

57


thonormal basis pe jq jPN of H such that Qe j δ je j, δ j ¥ 0, n P N. Moreover, for any

α α0 infnPN 1δn

we have

»H

eα

2 |x|2Nδ pdxq p8¹

j1

p1αδ jqq 12 pdetpIαQqq 1

2 .

Proof. The first assertion is obvious from the definition of Q P L1 pH q. Now since

Tr Q 8

j1

δ j 8

then it follows that

0 8¹

j1

p1αδ jq 12 8 for α α0.

Moreover,

»H

eα

2 |xx,e1y|2Nδ pdxq »R

eα

2 ξ 2Nδ1pdξ q 1?

2πδ1

»R

eα

2 ξ 2e

ξ 22δ1 dξ .

which is equal to p1δ1q 12 by some elementary computations. Therefore,

»H

eα

2 |x|2Nδ pdxq limnÑ8

»H

eα

2°n

j1 |xx,e jy|2Nδ pdxq

limnÑ8p

n¹j1

p1αδ jqq 12

pdetpIαQqq 12 .

by the monotone convergence theorem.

Proposition 3.3.6. We have

»H

exh,xyµpdxq exm,hy 12 xQh,hy, h PH

58


Proof. Let ε ¡ 0. Using the Cauchy-Schwarz inequality and an elementary inequality

that p?εx 1?εyq2 xy ¥ 0 for any x,y ¥ 0, respectively, we have

exh,xy ¤ ehx ¤ eεx2e

1εh2

Choosing ε 12α0 1

2 infnPN 1δn

, then the Lebesgue dominated convergence theorem

implies that

»H

exh,xyµpdxq limnÑ8

»H

exh,Pnxyµpdxq

limnÑ8

»H

ex1h1...xnhn µpdxq limnÑ8

n¹j1

»R

ex jh jNm j,δ jpdx jq

limnÑ8

n¹j1

e12 δ jh2

j limnÑ8e

12 xPnQh,hy

e12 xQh,hy.

59


CHAPTER 4

APPLICATIONS OF GAUSSIAN MEASURES

A. Random Variables on a Gaussian Hilbert Spaces

In this section we will use Gaussian measure to define and study the Gaussian random

variable. First, we will see that any affine mapping between Hilbert spaces, i.e., a

composition of a linear map with a translation, is indeed Gaussian.

Definition 4.1.1 (Gaussian random variable). A Gaussian random variable is a ran-

dom variable X such that its law is Gaussian. If X#P is Gaussian with mean m and

covariance Q, then we denote it by Nm,Q.

From the change of variables formula, we can conclude that the law of a random

variable X is Gaussian if and only if

yX#Pphq »

Ω

eixXpωq,hyPpdωq eixm,hy 12 xQh,hy.

Let H be a separable Hilbert space. We call H together with a Gaussian measure on

H as a Gaussian Hilbert space.

Proposition 4.1.1. Let µ Nm,Q be a Gaussian measure on pH ,BpH qq with mean m

and covariance Q. Let A P LpH ,K q where K is another Hilbert space and T pxq Ax b where x P H and b P K . If A is the transpose of A, then T is a Gaussian

random variable with mean Amb and covariance AQA.

Proof. The Fourier transform of the law of T is given by

yT#µpkq »

Keixk,xyT#µpdxq

»H

eixk,T pωqyµpdωq

»

Heixk,Aωby

µpdωq eixk,by»

HeixAk,ωy

µpdωq

eixk,Amby 12 xAQAk,ky NAmb,AQApkq.

60


for any vector k PK .

Corollary 4.1.1. Let µ Nm,δ be a Gaussian measure in H , then for any α α0 infnPN 1

δnwe have

»H

eα

2 |x|2Nm,δ pdxq pdetpIαQqq 12 e

α

2 xpIαQq1m,my.

Proof. By applying Proposition 4.1.1, we obtain

»H

eα

2 |x|2Nm,δ pdxq »

He

α

2 |xm|2Nδ pdxq

eα

2 |m|28¹

n1

1?2πδn

»R

eα

2 y2αmnyey2

2δn dy

eα

2 |m|28¹

n1

1?2πδn

»R

e

1αδn2δn

y2αmnydy

eα

2 |m|28¹

n1

1?2πδn

eδnα2m2

n2p1αδnq

»R

e1αδn

2δn

y δnαmn

1αδn

2

dy

8¹

n1

1?2πδn

eα

2 m2k e

δnα2m2n

2p1αδnq

»R

ey2dy

2δn

1αδn

12

8¹

n1

p1αδnq 12 e

αm2n

2p1αδnq

pdetpIαQqq 12 e

α

2 xpIαQq1m,my.

Corollary 4.1.2. Let µ NQ be a Gaussian measure on pH ,BpH qq with mean 0

and covariance Q. Take an arbitrary finite sequence z1, ...,zn PH and define a linear

map T : H Ñ Rn such that T pxq pxx,z1y , ...,xx,znyq for any x P H . Then T is an

Rn-valued Gaussian random variable with covariance Q, where pQqi, j @

Qzi,z jD

for

i, j 1, ...,n.

61


Proof. It is elementary to show that the transpose T : Rn ÑH of T is given by

T pαq n

k1

αkzk

for α pα1, ...,αnq P Rn. Therefore, we have

T QT pαq Tn

k1

αkpQzkq n

k1

αkpxQzk,z1y , ...,xQzk,znyq

and so the conclusion follows by Proposition 4.1.1.

Let H be a separable Hilbert space and X pX1, ...,Xnq be a H n-valued random

variable where Xi is a H -valued random variable for i 1, ...,n. Consider the Fourier

transform of Xi pi 1, ...,nq and X , i.e.,

Xiphq Ereixh,Xiys,h PH and Xphq Erei°n

i1xhi,Xiys,h ph1, ...,hnq PH .

Proposition 4.1.2. Let X pX1, ...,Xnq where X1, ...,Xn be H -valued random vari-

ables. Then the following three statements are equivalent

(i) X1, ...,Xn are independent.

(ii) If h ph1, ...,hnq PH n, then Xphq ±nj1 X jph jq.

(iii) If φ1, ...,φn are real bounded Borel functions, then

Erφ1pX1q, ...,φnpXnqs Erφ1pX1qs...ErφnpXnqs.

For complete details and proof, see Dudley, R.M.

Proposition 4.1.3. Let Xi Nmi,δi be a real and independent Gaussian random vari-

able with mean mi and covariance δi for i 1, ...,n. Then X pX1, ...,Xnq is an Rn-

62


valued Gaussian random variable with mean pm1, ...,mnq and covariance diagpδ1, ...,δnq,i.e., the diagonal of the matrix pδ1, ...,δnq.

B. Linear Random Variables on a Gaussian Hilbert Spaces

Among random variables in a separable Hilbert space H , of particular interest are

linear mappings from H into R. Let α PH , define a linear mapping F : H Ñ R as

follows

Fαpxq xα,xy , @x PH .

Since an inner product is a bounded linear operator, then Proposition 4.1.1 implies that

Fα is a Gaussian random variable. Furthermore, if α1, ...,αn PH , then

Fα1,...,αnpxq pFα1pxq, ...,Fαnpxqq, @x PH

is an Rn valued Gaussian random variable with mean 0 and covariance pQα1,...,αnqi, j @Qvi,v j

Dfor i, j 1, ...,n.

Proposition 4.2.1. Let α1, ...,αn PH , then Fα1, ...,Fαn are independent if and only if

its covariance matrix pQα1,...,αnq is diagonal, i.e.,

@Qαi,α j

D 0 if i j.

Proof. As usual, computing the Fourier transform of both Fαi and Fα1,...,αn , i 1, ...,n,

we obtain

xFαipxiq »

Heixxi,yyNxQαi,αiypdyq e

12 xxQαi,αiyxi,xiy e

12 x2

i xQαi,αiy

and similarly, Fα1,...,αnpx1, ...,xnq e12°n

i, j1pQα1,...,αnqi, jxix j

63


From Proposition 4.1.2, we know that Fα1, ...,Fαn are independent if and only if Fα1,...,αn ±ni1

xFαi , i.e., if and only if@

Qαi,α jD 0 if i j.

C. Equivalence Classes of Random Variables

Let pΩ,F ,Pq be a probability space and H be a separable Hilbert space.

Definition 4.3.1 (Equivalence between Hilbert space valued random variables). Let

E pH q be the collection of all H -valued random variables. We say that two random

variables X ,Y P E pH q are equivalent and denote it by X Y if

Pptω P Ω : Xpωq Y pωquq 1,

i.e., they only differ on a null set.

It is easy to check that the relation on E pH q is an equivalence relation which

implies that the set E pH q is a disjoint union of equivalence classes.

Proposition 4.3.1. Every random variables in the same equivalence class rX have the

same law, which is called the law of rX.

Proof. Let X ,Y P rX be two equivalent H -valued random variables, then for any I PBpH q we have

PpX P Iq PpX P I,X Y qPpX P I,X Y q PpY P I,X Y q ¤ PpY P Iq

In a similar way one can clearly see that PpY P Iq ¤ PpX P Iq. Thus we have pX#PqpIq PpX1pIqq PpY1pIqq pY#PqpIq for all I P BpH q and the conclusion follows.

From now on, we will not worry about the distinction between a random variable

and its equivalence class, except when the precise pointwise values of a representative

function are significant.

64


Definition 4.3.2. Let 1¤ p 8. The space LppΩ,F ,P;H q consists of equivalence

classes of random variables X : Ω ÑH such that³

Ω|Xpωq|pPpdωq 8. The Lp-

norm of X P LppΩ,F ,P;H q is defined by XLppΩ,F ,P;H q ³

Ω|Xpωq|pPpdωq 1

p .

Lemma 4.3.1. Let 1 ¤ p 8 and tXn P LppΩ,F ,P;H q : n P Nu be a sequence of

Lp-random variables such that°8

n1 XnLppΩ,F ,P;H q 8, then the series°8

n1 Xn

converges pointwise to some random variable X P LppΩ,F ,P;H q.

Proof. Let M°8i1 XnLppΩ,F ,P;H q and define Yn,Y : ΩÑr0,8s by Yn

°ni1 |Xn|

and Y °8i1 |Xn|, then Yn ÑY pointwise. Hence we have limnÑ8

³Ω

Y pn pωqPpdωq³

ΩY ppωqPpdωq by the monotone convergence theorem. Moreover, by the Minkowski

inequality, we also have YnLppΩ,F ,P;H q¤°n

i1 XnLppΩ,F ,P;H q¤M for every n PN.

It follows that Y P LppΩ,F ,P;H q and Y LppΩ,F ,P;H q ¤ M. We also have that the

sum°8

n1 Xn is absolutely convergent pointwise a.e., so it converges pointwise to a

function X P LppΩ,F ,P;H q with |X | ¤ Y . Since |X °ni1 Xi|p Ñ 0 as n Ñ 8

and is dominated by an integrable function ,i.e., |X °ni1 Xi|p ¤ p|X |°n

i1 |Xi|qp ¤p2Y qp, the Lebesgue dominated convergence theorem implies that limnÑ8

³Ω|X °n

i1 Xi|pPpdωq 0. This means that°8

n1 Xn converges to X in LppΩ,F ,P;H q and

so the conclusion follows.

The following theorem shows that LppΩ,F ,P;H q equipped with the Lp-norm is

a Banach space.

Theorem 4.3.1 (Riesz-Fischer theorem). Let 1¤ p 8, then LppΩ,F ,P;H q equip-

ped with the Lp-norm is complete.

Proof. Let pXnqnPN be a Cauchy sequence in LppΩ,F ,P;H q. Take a subsequence

pXn jq jPN such that Xn j1 Xn jLppΩ,F ,P;H q ¤ 12 j for every j P N. By letting Yj

Xn j1 Xn j , we have°8

j1 YjLppΩ,F ,P;H q 8 and so by lemma 4.3.1 the sum

Xn1°8

j1Yj converges pointwise a.e. to a function X P LppΩ,F ,P;H q. This implies

65


that the limit of the subsequence pXn jq exist in LppΩ,F ,P;H q since

limjÑ8

Xn j limjÑ8

Xn1

j1

i1

Yj

Xn1

8

j1

Yj X .

Now it is elementary to see that since pXnqnPN is Cauchy, then limnÑ8Xn X .

Now we shall show that L2 limits of sequences of Gaussian random variables is

Gaussian.

Proposition 4.3.2. Let pXnqnPN be a sequence of H -valued Gaussian random vari-

ables with mean mn and covariance Qn such that Xn Ñ X in L2pΩ,F ,P;H q. If m and

Q are the mean and the covariance respectively of the law of X, then limnÑ8 xmn,hyxm,hy and limnÑ8 xQnh,hy xQh,hy for every h P H . Furthermore, we also have

that X is Gaussian.

Proof. Let h PH . Since Xn Ñ X in L2pΩ,F ,P;H q, we have

limnÑ8xmn,hy lim

nÑ8

»Ω

xXnpωq,hyPpdωq »

Ω

xXpωq,hyPpdωq xm,hy

and

limnÑ8xQnh,hy lim

nÑ8

»Ω

xXnpωqmn,hy2Ppdωq

»

Ω

xXpωqm,hy2Ppdωq

xQh,hy .

Finally, we will show that X is Gaussian Nm,Q via its Fourier transform and by using

the change of variables formula as follows

»H

eixx,hyX#Ppdxq »

Ω

eixXpωq,hyPpdωq limnÑ8

»Ω

eixXnpωq,hyPpdωq

66


limnÑ8eixmn,hy 1

2 xQnh,hy

eixm,hy 12 xQh,hy.

D. The Cameron-Martin Space and The White Noise Mapping

Let H be a separable Hilbert space and µ NQ be a nondegenerate Gaussian measure.

It is possible to associate to µ in a canonical way a Hilbert space Hµ H , called the

Cameron-Martin space of µ . One of the most important properties in this space is

that characterises precisely those directions in H in which the translations leave the

measure µ quasi-invariant in the sense that the translated measure has the same null

sets as the original measure. Moreover, the space Hµ will always be strictly smaller

than H since dimpHµq 8 and even µpHµq 0. Contrast this to the case of finite

dimensional Lebesgue measure which is invariant under translation in any direction.

This is a striking illustration of the fact that measures in infinite dimensional spaces

have a strong tendency of being mutually singular, i.e., for any two infinite dimensional

measures µ and v on H , there exist two disjoint measurable sets A and B such that

its union is H and µ is zero for any measurable subsets of A while v is zero for any

measurable subsets of B. Recall that since Q is symmetric and compact, then there exist

a complete orthonormal basis pe jq jPN in H such that Qe j δ je j where pδ jq jPN are the

eigenvalues of Q. Also, since the inverse of the covariance Q is given by Q1e j 1δ j

e j

for every j P N where δ j Ñ 0 as n Ñ 8 since Q has finite trace, then Q1 is not

continuous. Hence, it is a direct consequences of the closed graph theorem that the

range QpH q does not coincide with H . However the following lemma shows that

QpH q is dense in H .

Lemma 4.4.1. QpH q is dense in H .

67


Proof. It is enough to show that the orthogonal complement of QpH q is the trivial

space. It is clear from the continuity of an inner product that QpH qK QpH qK and

hence QpH q QpH qKKQpH qKK by the closedness of a closure. So, if QpH qKt0u, then QpH q QpH qKK H , i.e., QpH q is dense in H . Take a vector x PH

such that xQy,xy 0 for every y PH . Therefore, we have xy,Qxy 0 so that Qx 0

and hence x 0 since µ is nondegenerate.

If we denote by xn xx,eny, then we have

Qx 8

n1

δnxnen and Q12 x

8

n1

δ12

n xnen.

We set also the corresponding finite dimensional transformation

Qnx n

i1

δixiei and Q12n x

n

i1

δ12

i xiei

and call the space Q12 pH q as the Cameron-Martin space of µ . In a similar way as

before, Q12 pH q is a proper subspace of H which is dense in H . Now let α,x P H

and consider the linear functionals Gα,npxq B

x,Q 1

2n α

F °n

i1 δ 1

2i xiαi. If α P

Q12 pH q, then one can define the function Gαpxq

Ax,Q 1

2 α

E °8

i1 δ 1

2i xiαi for

x PH . Indeed, this mapping is linear since for any c1,c2 PR, we have Gαpc1xc2yq c1Gαpxq c2Gαpyq.

Proposition 4.4.1. We have µpQ 12 pH qq 0.

Proof. We set

An $&%y PH :

8

j1

δ1j y2

j n2

,.- and An,k $&%y PH :

2k

j1

δ1j y2

j n2

,.-for any n,k P N. It is obvious from the definition of An and An,k that An Ò Q

12 pH q

68


as n Ñ 8 and An,k Ó An as k Ñ 8. Therefore, we need to show that µpAnq limkÑ8 µpAn,kq 0 as follows

µpAn,kq »tyPRk:

°2kj1 δ

1j y2

j n2u

2k¡j1

Nδ jpdy jq »tzPR2k:|z| nu

NI2kpdzq

where zk δ 1

2k yk and I2k is the identity in R2k. We have

µpAn,kq µpAn,kqµpH q

³n0 e

r22 r2k1dr³8

0 er22 r2k1dr

³ n2

20 essk1ds³80 essk1ds

.

Therefore,

µpAn,kq 1pk1q!

» n22

0essk1ds ¤ 1

pk1q!» n2

2

0sk1ds 1

k!

n2

2

k


Recall the mapping G given by

$''&''%G : Q

12 pH q H Ñ L2pH ,µq, α ÞÑ Gα

Gαpxq A

x,Q 12 α

E, x PH

,

then clearly G is an isomorphism of Q12 pH q into L2pH ,µq because

»H

Gα1pxqGα2pxqµpdxq A

QQ 12 α1,Q 1

2 α2

E xα1,α2y .

Therefore, since Q12 pH q is dense in H , then the mapping G can be uniquely extended

to the whole H via the continuous extension theorem. Thus for any α P Q12 pH q,Gα

is a well-defined element of L2pH ,µq. From now on, we will call Gα as the white

noise mapping.

69


Proposition 4.4.2. Let µ NQ be a Gaussian measure on H , then the limit limnÑ8

Gα,n Gα exist in the space of all measurable functions

L2pH ,µq "

f : H Ñ R :»

H| f pxq|2µpdxq 8

*.

Furthermore, for α PH , we have

»H|Gαpxq|2µpdxq |α |2.

Proof. We have

»H|Gα,nmpxqGα,npxq|2µpdxq

»H|

nm

in1

δ 1

2i xiαi|2µpdxq

nm

i, jn1

pδiδ jq 12 αiα j

»H

xix jµpdxq

nm

in1

δ1i α

2i

»H

x2i µpdxq

np

in1

α2i

by the orthogonality of e j. Therefore, we conclude that the sequence pGα,nqnPN is

Cauchy in L2pH ,µq. For the second statement, we have

»H|Gαpxq|2µpdxq lim

nÑ8

»H|Gα,npxq|2µpdxq lim

nÑ8

n

i1

δ1i α

2i

»H

x2i µpdxq

limnÑ8

n

i1

α2i

|α |2

by the Lebesgue dominated convergence theorem.

Remark 4.4.1. The above proposition shows that the series Gαpxq°8

i1 δ12

i xx,eiyx f ,eiyis being convergent on L2pH ,µq. By the same arguments as in the proof of proposition

70


4.4.2 one can show that Gα is well-defined as an L2pH ,µq function.

Proposition 4.4.3. Let α PH , then³H eGα pxqµpdxq e

12 |α|2 .

Proof. We have,

»H

eGα pxqµpdxq limnÑ8

»H

eGα,npxqµpdxq limnÑ8

»H

e

Bx,Q

12

n α

Fµpdxq

limnÑ8e

12 |αn|2

e12 |α|2.

Proposition 4.4.4. Let f be a mapping from H into L2pH ,µq given by α Ñ eGα

,

then f is continuous.

Proof. Since

»H

eGα eGβ

2NQpdxq

»H

e2Gα 2eGαβ e2Gβ

NQpdxq

e2|α|2 2e12 |αβ |2 e2|β |2

e|α|2 e|β |

222e|α|

2|β |2

1 e12 |αβ |2

,

then for any convergent sequence αn Ñα , we have limnÑ8³H

eGαn eGα

2 NQpdxq limnÑ8

e|αn|2 e|α|2

22e|αn|2|α|2

1 e

12 |αnα|2

0. This shows the con-

tinuity in α of eGα .

Proposition 4.4.5. Let µ NQ be a Gaussian measure on H , then the limit limnÑ8 eGα,n

eGα exist in L2pH ,µq for a given α PH .

Proof. By using the fact that³H exα,xyNQpdxq e

12 xQh,hy , we obtain

»H|eGα,n eGα,m |2µpdxq

71


»

H

e2B

α,Q 1

2n x

F2e

Bα,Q

12

n xFB

α,Q 1

2m x

F e

2B

α,Q 1

2m x

F µpdxq

e2°n

i1 α2i e2

°mi1 α2

i 2»

He

BpQ 1

2n Q

12

m qα,xF

µpdxq

e2°n

i1 α2i e2

°mi1 α2

i 2e2°n

i1 α2i 1

2°m

in1 α2i

e2°n

i1 α2i

1 e2

°min1 α2

i 2e12°m

in1 α2i

which is converges to zero as n,m Ñ 8. This proves thateGα,n

is a Cauchy se-

quence in L2pH ,µq.

Proposition 4.4.6. Gα is a real Gaussian random variable N|α|2 .

Proof. We need to show that the Fourier transform of Gα is given by

pGαq#µphq »R

eihypGαq#µpdyq »

HeihGα pxqµpdxq e

12 h2|α|2 .

Take a sequence pαnqnPN in Q12 pH q such that αn Ñ α in H as n Ñ8. Then, we

have

»H

eihGα pxqµpdxq limnÑ8

»H

eihB

Q 12 αn,x

Fµpdxq lim

nÑ8e12 h2|αn|2 e

12 h2|α|2

by the Lebesgue dominated convergence theorem and the conclusion follows.

Proposition 4.4.7. Let α1, ...,αn P H and pGα1, ...,Gαnq be an Rn-valued random

variable. Then the law of pGα1 , ...,Gαnq is given by

pGα1 , ...,Gαnq#P NQ

where Qi, j @

αi,α jD

for i, j 1, ...,n. Furthermore, the random variables Gα1, ...,Gαn

are independent if and only if α1, ...,αn form an orthogonal set.

72


Proof. Recall the linear mappings from H into R given by Fαpxq xα,xy, then we

have for any x P H ,Gαi FQ1z2αipxq pi 1, ...,nq. Then Corollary 4.1.2 implies the

first statement and Proposition 4.2.1 implies the second one.

Corollary 4.4.1. The random variables Gα1, ...,Gαn are independent if and only if

α1, ...,αn are independent.

Proof. This is a direct consequences of Proposition 4.2.1 for Fαi FQ12αi

E. The Malliavin Derivative

Let µ NQ be a nondegenerate Gaussian measure on a separable Hilbert space H .

Let pe jq jPN and pδ jq jPN be an orthonormal basis in H and a sequence of positive

numbers, respectively, such that Qe j δ je j. Define the orthogonal projection operator

Pn : H Ñ PnpH q by Pnx °ni1 xx,eiyei. For any k P N, we denote by Ck

bpH q the

subspace of CbpH q of all mappings in CbpH q with their Frechét derivatives of order

less or equal to k together with the norm f k f 0°k

n1 supxPH |Dh f pxq| for any

f PCkbpH q. For any x PH and j P N, we set x j

@x,e j

Dand for any f PC1

bpH q, we

define the derivative of f in the direction of e j by

D j f @D f ,e j

D limcÑ0

1cr f px ce jq f pxqs, x PH .

Proposition 4.5.1. CkbpH q is a Banach space.

We denote by L2pH ,µq the Hilbert space of all equivalence classes of µ-square

integrable functions f : H Ñ R equipped with the inner product x f ,gyL2pH ,µq ³H f gdµ . This inner product clearly induces the norm f L2pH ,µq

³H | f pxq|2µpdxq 1

2 .

Moreover, we will also denote by L2pH ,µ;H q the Hilbert space of all equivalence

classes of µ-square integrable functions f : H Ñ H equipped with the inner prod-

uct x f ,gyL2pH ,µ;H q ³H x f pxq,gpxqyµpdxq for any f ,g P L2pH ,µ;H q. Clearly, the

corresponding norm is given by f L2pH ,µ;H q ³

H | f pxq|2µpdxq 12 .

73


Proposition 4.5.2. Let f P L2pH ,µ;H q and f j @

f ,e jD

for any j PN. Then for any

j PN, we have f j P L2pH ,µq. Furthermore, the series°n

j1 f jpxqe j converges to f pxqin L2pH ,µ;H q.

Proof. These results are direct consequences of the Cauchy-Schwarz inequality and

the Lebesgue dominated convergence theorem, respectively.

Definition 4.5.1 (Exponential space). The space EpH q of linear span in CbpH q of

all real and imaginary parts of functions ξh, h PH , where ξhpxq eixx,hy for x PH is

called the space of exponential functions.

Now consider the linear operator MQ12 D : EpH q L2pH ,µqÑ L2pH ,µ;H q.

This operator is well-defined for all α P H whereas DGα is only defined for α PQ

12 pH q H . This fact is essential in the study of Brownian motion and stochastic

differetial equations. We shall show that M is closable and the domain of its clo-

sure (still denoted by M) will be called the Malliavin-Sobolev space and denoted by

D1,2pH ,µq.

F. Approximation by Exponential Functions

The exponential space EpH q is not dense in CbpH q. However, the following expo-

nential approximation result holds.

Proposition 4.6.1. Let f P CbpH q. Then there exist a two-index sequence p fk,nq in

EpH q such that for any x PH , limk,nÑ8 fk,npxq f pxq and fk,n0 ¤ f 0.

Proof. Using the fact that a closed ball is compact if and only if the normed space is

finite dimensional, we divide the proof into two cases as follows

(i) Assume that H is finite dimensional. Let f PCbpH q, ε ¡ 0, and k PN be fixed.

By the Stone-Weierstrass theorem, there exist a sequence p fk,nqnPN in EpH qsuch that | f pxq fk,npxq| ε for any x in the closed ball with center 0 and radius

74


k, Bkp0q. Therefore, we have fk,n0 ¤ f 0ε and since ε ¡ 0 is arbitrary, then

the conclusion follows by a standard diagonal extraction argument.

(ii) Assume that H is infinite dimensional and pe jq jPN the corresponding complete

orthonormal system such that Pkpxq °k

j1@

x,e jD

e j, x P H . Let f P CbpH qand set fkpxq f pPkxq for every x P H and k P N. Let k be fixed, from the

finite dimensional case, there exist a sequence p fk,nqnPN in EpH q such that

limnÑ8 fk,npxq fkpxq for every x P H and fk,n0 ¤ fk0 ¤ f 0. There-

fore, for every x P H , we have limk,nÑ8 fk,npxq limkÑ8 fkpxq f pxq and

the conclusion follows.

Clearly we can generalize the above result to the following proposition.

Proposition 4.6.2. Let f P C1bpH q. Then there exist a two-index sequence p fk,nq

in EpH q such that for any x P H , limk,nÑ8 fk,npxq f pxq, limk,nÑ8D fk,npxq D f pxq, and fk,n0D fk,n0 ¤ f 0D f 0.

Proof. The proof is similar as the previous proposition since C1bpH q is a subspace of

CbpH q.

Corollary 4.6.1. The exponential space EpH q is dense in the space of all square

integrable function L2pH ,µq.

Proof. The space CbpH q is dense in L2pH ,µq by using the Dynkin π λ theorem.

Clearly Proposition 4.6.1 shows that EpH q is dense in CbpH q, and the conclusion

follows.

Let EpH ,H q denote the linear span of all functions of the form f pxq ξhpxqz for

every x,z PH . Then we also have the following proposition.

Proposition 4.6.3. The space EpH ,H q is dense in the space L2pH ,µ;H q of all

µ-square integrable mappings on H .

75


G. The Malliavin-Sobolev Space D1,2pH ,µq

In this section we shall prove that the linear operator M Q12 D defined on the expo-

nential space EpH q is closable.

Lemma 4.7.1. Let f ,g P EpH q, then the following identity holds

»HpD j f qgdµ

»H

f pD jgqdµ 1δk

»H

x j f gdµ

for every j P N.

Proof. In view of corollary 4.6.1, it is enough to prove the identity above for any

functions f ξh1 and g ξh2 , h1,h2 P H . In this case, by the definition of Dk, we

have

»HpD j f qgdµ i f j

»H

eixx, fgydµ i f je12 xQp fgq, fgy (4.1)

and

»H

f pD jgqdµ ig j

»H

eixx, fgydµ ig je12 xQp fgq, fgy. (4.2)

Furthermore, we also have

»H

x j f gdµ »

Hx jeix fg,xydµ i

ddt

»H

eix fgte j,xydµ

t0

iddtre 1

2xQp fgte jq, fgte jyst0

i@

Qp f gq,e jD

e12 xQp fgq, fgy

iδ jp f jg jqe 12 xQp fgq, fgy. (4.3)

The equalities p3q,p4q, and 5 yields the desired identity.

76


Corollary 4.7.1. Let f ,g P EpH q, and α PQ12 pH q. Then the following identity holds

»HxM f ,αygdµ

»HxMg,αy f dµ

»H

Gα f gdµ.

Proof. By lemma 4.7.1 we have

»HxM f ,αygdµ

8

j1

δ jα j

»HpD j f qgdµ

8

j1

δ12j α j

»HpD jgq f dµ

8

j1

»H

δ 1

2j α jx j f gdµ

»

HxMg,αy f dµ

»H

Gα f gdµ

as desired.

Proposition 4.7.1. The mapping M : EpH q L2pH ,µq Ñ L2pH ,µ;H q defined by

M f Q12 D f is closable.

Proof. Take an arbitrary sequence p fnqnPN in EpH q such that fn Ñ 0 in L2pH ,µq and

M fn Ñ F in L2pH ,µ;H q as n Ñ8. To show that M is closable, we need to prove

that F 0. Let g P EpH q and α P Q12 pH q. Then by corollary 4.7.1 we have

»HxF,αygdµ lim

nÑ8

»HxM fn,αygdµ

»HxMg,αy fndµ

»H

Gα fngdµ 0

by using the fact that fn Ñ 0. But since g P EpH q is arbitrary, then by substituting

g xF,αy, we have³H xF,αy2 dµ 0 and consequently xF,αy. Again, by the arbi-

trariness of α and the denseness of Q12 pH q in H , we have F 0 as desired.

From now on, we shall denote the closure Q12 D by M and by D1,2pH ,µq the

domain of M. The space D1,2pH ,µq endowed by the scalar product x f ,gyD1,2pH ,µq ³H p f gxM f ,Mgyqdµ for every f ,g P D1,2pH ,µq is a Hilbert space.

77


Definition 4.7.1 (Malliavin derivative). Let f PD1,2pH ,µq, then the Malliavin deriva-

tive of f is given by M f .

Now we are going to prove some useful properties of the space D1,2pH ,µq. The

following proposition shows that every function of class C1 is belong to the space

D1,2pH ,µq.

Proposition 4.7.2. Let f PC1bpH q, then f PD1,2pH ,µq and for every x PH we have

M f pxq Q1,2D f pxq µ-a.e.

Proof. Let f PC1bpH q. In view of Proposition 4.6.2, there exist a two index sequence

p fk,nq in EpH q such that limk,nÑ8 fk,n f and limk,nÑ8D fk,n D f . Then, the

Lebesgue dominated converence theorem implies that

limk,nÑ8

»H| fk,n f |2dµ Ñ 0 and lim

k,nÑ8

»H|D fk,nD f |2dµ Ñ 0,

i.e., limk,nÑ8 fk,n f in L2pH ,µq and limk,nÑ8D fk,n D f in L2pH ,µ;H q.Consequently, f PD1,2pH q by the closedness of D1,2pH q and M f Q

12 D f µ-a.e.

Proposition 4.7.3. Let f : H Ñ R be a continuously differentiable function and there

exist a nonnegative integer N and c¡ 0 such that | f pxq||D f pxq| ¤ cp1|x|2Nq. Then

f P D1,2pH ,µq and M f pxq Q12 D f pxq.

Proof. For the case that N 0 is similar as above proposition. Now let N P N and set

fnpxq f pxq1n1|x|2n for every x PH . Since

| fnpxq f pxq| 1n

|x|2N

1n1|x|2n ¤cnp1|x|2Nq|x|2N .

Then we have,

limnÑ8

»H| fnpxq f pxq| ¤ lim

nÑ8c2

n2

»Hp1|x|2Nq2|x|4Ndµ lim

nÑ8c0

n2 0

78


for some c0 P R. This shows that limk,nÑ8 fn f in L2pH ,µq. In a similar way, we

can show that limnÑ8M fn M f in L2pH ,µ;H q, and the conclusion follows.

H. Brownian Motion in Gaussian Hilbert Space

Definition 4.8.1 (Real Stochastic Process). Let pΩ,F q be a measurable space and

T ¡ 0 be a fixed positive number. A real stochastic process X on r0,T s is a measurable

function of two variables X : r0,T s Ω Ñ R such that X1pIq P Bpr0,T sq F for

every I P BpRq.

For a fixed instant of time t0, the F -measurable mapping Xpt0,ωq Xt0pωq is a R-

valued random variable. For a fixed random outcome ω0 PΩ, the Bpr0,T sq-measurable

mapping Xpt,ω0q Xω0ptq is a function of time, we call this function a realization or

a sample path of the stochastic process X .

Definition 4.8.2 (Equivalence of Stochastic Processes). Two stochastic processes X

and Y defined on the product space r0,T sΩ are equivalent if for every t P r0,T s, we

have

Pptω P Ω : Xpt,ωq Y pt,ωquq 1,

i.e., they only differ on a null set.

The equivalence of stochastic processes is an equivalence relation. Hence, we can

consider the equivalence classes of stochastic processes. We call any element in the

equivalence class of processes X as the version of X , i.e., a stochastic process Y pt,ωq is

said to be a version of Xpt,ωq if Pptω P Ω : Xpt,ωq Y pt,ωquq 1 for any t P r0,T s.

Definition 4.8.3 (Continuous Stochastic Processes). A stochastic process X : r0,T sΩÑR is said to be continuous if its sample paths are continuous for P-almost all ω PΩ. An equivalence class of processes X is said to be continuous if it has a continuous

version.

79


On the other hand, for p ¥ 1 a process X is said to be p-mean continuous if

Ep|Xt |pq 8 for any t P r0,T s and the mapping r0,T s Ñ LppΩ,Pq given by t ÞÑ Xt is

continuous. An equivalence class of processes X is said to be p-mean continuous if all

of its version is p-mean continuous. Moreover, we denote by Cpr0,T s;LppΩ,F ,Pqqthe Banach space of all p-mean continuous equivalence class of processes endowed

with the norm X psuptPr0,T sEp|Xt |pqq1p .

Definition 4.8.4 (Gaussian Process). A stochastic process X : r0,T sΩÑR is said to

be Gaussian if for every t1, ..., tn P r0,T s such that t1 ... tn, we have the Rn-valued

random variable pXt1, ...,Xtnq is Gaussian.

Definition 4.8.5 (Real Brownian Motion). A continuous real stochastic process B is

called a (standard) Brownian motion or a Wiener process on r0,T s if it satisfies the

following conditions:

(i) It starts at zero: B0 0.

(ii) For every s, t P r0,T s such that s t, Bt Bs is a real Gaussian random variable

with law Nts.

(iii) It has independent increments: For every t1, ..., tn P p0,T s such that t1 ... tn,

the random variables

Bt1,Bt2 Bt1, ...,Btn Btn1

are independent.

We say that an equivalence class of stochastic processes X a Brownian motion if it

has a version which is a Brownian motion. In the following proposition, we show that

any Brownian motion is a Gaussian process.

Proposition 4.8.1. Let B with t P r0,T s be a Brownian motion on a probability space

pΩ,F ,Pq. Then B is a Gaussian process. Spesifically, for any t1, ..., tn P p0,T s such

80


that t1 ... tn, we have pBt1, ...Btnq NQ where the covariance operator Q is given

by Qt1,...,tn SDt1,...,tnS. Here, Dt1,...,tn diagpt1, t2 t1, ..., tn tn1q, S P LpRnq is

defined by Spx1, ...,xnq px1,x1 x2, ...,x1 x2 ... xnq, and S is the adjoint of S.

Furthermore, for any A P BpRnq, we have PppBt1, ...,Btnq P Aq is equal to

1ap2πqnt1pt2 t1q...ptn tn1q

»A

e x2

12t1 px2x1q

2

2pt2t1q... pxnxn1q

2

2ptntn1q dx1...dxn.

Proof. Let t1, ..., tn P p0,T s such that t1 ... tn. Set X pBt1 ,Bt2Bt1 , ...,BtnBtn1qand Y pBt1, ...,Btnq. Since Bt is a Brownian motion, then the increments Bt1,Bt2 Bt1, ...,Btn Btn1 are independent and hence, in view of Proposition 4.1.3, X is an n-

dimensional Gaussian random variable NQ with Qx diagpt1, t2 t1, ..., tn tn1q Dt1,...,tn . Now let S P LpRnq such that Spx1, ...,xnq px1,x1 x2, ...,x1 x2 ... xnq,then we have rSpXqsk

°ki1pBtiBti1q Btk B0 Btk so that SpXq Y . Therefore,

by Proposition 4.1.1, we conclude that Y pBt1 , ...,Btnq is Gaussian with mean 0 and

covariance Q given by Qt1,...,tn SDt1,...,tnS where S is the adjoin of S. Next, we will

show the identity for PppBt1, ...,Btnq P Aq. Notice that

detpSq det

1 1 1 . . . 1

0 1 1 . . . 1

. . . . . . . . . . . . . .

0 0 0 . . . 1

n¹i1

rSsii 1.

It is also easy to compute that S1 P LpRnq is given by S1pxq px1,x2 x1, ...,xnxn1q. Therefore, using the density of multivariate Gaussian distribution, we have

PppBt1,Bt2, ...,Btnq P Aq 1ap2πqn detpSDt1,...,tnSq»

Ae

12xpSDt1,...,tnSq1x,xydx1...dxn

81


Since

@pSDt1,...,tnSq1x,xD @ppSq1D1

t1,...,tnS1qx,xD @ppS1qD1

t1,...,tnS1qx,xD @pD1

t1,...,tnS1qx,S1xD

x21

t1 px2 x1q2

t2 t1 ... pxn xn1q2

tn tn1.

Clearly, we have detpDt1,...,tnq t1pt2t1q...ptntn1q so that detpSDt1,...,tnSq t1pt2t1q...ptn tn1q and the conlusion follows.

Now we will construct Brownian motion B Bt , t P r0,T s, in a probability space

pH ,BpH q,µq where H L2p0,T q and µ NQ for Q P L1 pH q such that kerQ t0u.

Lemma 4.8.1. Let m¡ 1, α P p 12m ,1q, T ¡ 0, and f P L2mp0,T ;H q. For any t P r0,T s,

set Fptq ³t0ptσqα1 f pσqdσ . Then F PCpr0,T s;H q.

Proof. First, notice that 2mα 1 ¡ 0. By Holder’s inequality, we have

|Fptq| ¤» t

0ptσqpα1q 2m

2m1 dσ

2m12m

| f |L2mp0,T ;H q.

Therefore, F P L8p0,T ;H q and F is continuous at 0. We shall prove that F is contin-

uous on r t02 ,T s for any t0 P p0,T s. We set for ε t0

2 , Fεptq ³tε

0 ptσqα1 f pσqdσ . It

is obvious that Fε is continuous on r t02 ,T s. Moreover, using Holder inequality, we also

have

|FptqFεptq| ¤ M

2m12mα 1

2m12m

εα 1

2m | f |L2mp0,T ;H q.

Thus, Fε Ñ F as ε Ñ 0, uniformly on r t02 ,T s, and the conclusion follows.

Theorem 4.8.1. Let T ¡ 0 and 1r0,ts be the indicator function of r0, ts for t P r0,T s. Let

Bt be the equivalence class of processes defined by Bt G1r0,ts where G is the white-

82


noise mapping. Then B is a Brownian motion. Furhermore, B PCpr0,T s,L2mpH ,µqqfor any m P N.

Proof. In view of Proposition 4.4.6, it is clear that for any t P r0,T s, Bt is a Gaus-

sian random variable Nt and for any 0 ¤ s t, Bt Bs W1rs,ts is a Gaussian random

variable Nts. Now notice that

1r0,t1s,1pt1,t2s, ...,1ptn1,tns(

form an orthogonal set of

elements of H . Hence, it follows from Proposition 4.4.7 that the increments Bt1,Bt2 Bt1, ...,Btn Btn1 are independent. Next, to show that B P Cpr0,T s,L2mpH ,µqq, we

notice that, since Bt Bs is Gaussian random variable Nts for every t ¡ s, then

Ep|Bt Bs|2mq ³R |x|2mNtspdxq p2mq!

2mm! pt sqm. Hence, B P Cpr0,T s,L2mpH ,µqqfor every m P N. Finally, we will show the continuity of the sample path. We shall

use the factorization method based on the following identity for 0 ¤ s ¤ σ ¤ t and

α P p0,1q :

» t

sptσqα1pσ sqαdσ

» 1

0p1 xqα1xαdx π

sinpπαq ,

where σ xpt sq s. The last equality is obtained based on the properties of beta

function. Let α P p0, 12q. Using the identity above, we have for s ¥ 0

1r0,tspsq sinpπαq

π

» t

0ptσqα1hσ psqdσ

where hσ psq 1r0,σ spsqpσ sqα . Moreover, hσ P L2p0,T q and |hσ |2 σ12α

12αsince

α P p0, 12q. Since the mapping H Ñ L2pH ,µq given by α ÞÑ Gα is continuous, we

obtain the following representation of B,

Bt sinpπαqπ

» t

0ptσqα1Ghσ

dσ .

Now using the fact that Ghσis a real Gaussian random variable with law Nσ12α

12α

, we

83


have »H|Ghα

pxq|2mµpdxq p2mq!

2mmp12αqm

σmp12αq.

Since α P p0, 12q, the Fubini theorem implies that

» T

0

»H

|Ghαpxq|2m

µpdxqdσ »

H

» T

0|Ghα

pxq|2mdσ

µpdxq 8.

Thus, the mapping σ ÞÑ Ghσbelongs to L2mp0,T q, µ-a.e., and the conclusion follows.

84


CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

A. Conclusions

We have discussed a Gaussian analysis in an infinite dimensional separable Hilbert

spaces. Since Lebesgue measure cannot be extended to infinite dimensional case, a nat-

ural substitute is given by Gaussian measure. To build the existence of such measure,

we use the trace class operator as the covariance function. We have constructed such

a Gaussian measure starting from the finite dimensional case in R to Rn via product

of n-Gaussian measures in R. Moreover, in an arbitrary infinite dimensional separable

Hilbert spaces, we first constructed a measure in R8, the space of all real sequences

via infinite product of Gaussian measures in R. Using the fact that such a measure is

concentrated on the space of all square summable real sequences l2-space, we show

that it is a Gaussian measure in the l2-space, and hence in the infinite dimensional sep-

arable Hilbert space H via the isomorphism of both spaces. An important application

of these results is in the context of stochastic analysis.

Next, we use Gaussian measures to define and study some important properties of a

Gaussian random variables in a separable Hilbert space. For example, we have shown

that any linear and affine transformation between separable Hilbert spaces is a Gaus-

sian random variable. We also define the white-noise mapping Gα for α P Q12 pH q,

i.e., the Cameron-Martin space of µ , which is Gaussian. Moreover, G transform pair-

wise orthogonal elements α1, ...,αn of H into independent Gaussian random variables

Gα1, ...,Gαn . White-noise mapping have a fundamental role in constructing Brownian

motion B Bt , t P r0,T s on the probability space pH ,BpH q,µq for H L2p0,T qand µ is a Gaussian measure with covariance Q . It clearly shows that the Brownian

motion Bt defined by Bt G1r0,ts , t P r0,T s is a Gaussian real random variable with

independent increments. We also study the Malliavin derivative M Q12 D defined

85


on the space of all exponential functions. We have shown that M is closable in the

space L2pH ,µq. We call the domain of its closure as the Malliavin-Sobolev space

D1,2pH ,µq which contains every function of class C1.

B. Recommendations

In this thesis, we only consider the construction of Gaussian measures in infinite sep-

arable Hilbert spaces via trace class operator as covariance function. It is possible

both to extend the state space (e.g. to a Banach spaces) or to consider another method

by retaining the identity operator instead of trace class operator with some cost. It is

also recommended to study further applications of the topics we have discussed here

such as Wiener integral, multidimensional Brownian motion, and its applications to

differential equations.

86


BIBLIOGRAPHY

Da Prato, G. 2006. An Introduction to Infinite Dimensional Analysis. Heidelberg:

Springer.

Da Prato, G. 2014. Introduction to Stochastic Analysis and Malliavin Calculus. Hei-

delberg: Springer.

Dudley, R.M. 2004. Real Analysis and Probability. Cambridge: Cambridge University

Press.

Mikosch, T. 1998. Elementary Stochastic Calculus with Finance in View. Singapore:

World Scientific.

Oden, J.T. and Demkowicz, L.F. 2010. Applied Functional Analysis, 2nd ed. Boca-

Raton: CRC Press.

Rudin, W. 1976. Principles of Mathematical Analysis, 3rd ed. New York: McGraw-

Hill.

Suryawan, H.P. 2014. A Problem on Measures in Infinite Dimensional Spaces, Prosi-

ding Seminar Nasional Matematika, Statistika, Pendidikan Matematika, dan

Komputasi. Surakarta: Universitas Negeri Sebelas Maret.

Tao, T. 2011. An Introduction to Measure Theory, Vol 126. Providence: American

Mathematical Society.

Wang, A. 2011. Lebesgue Measure and L2 space. Chicago: Chicago University.

Yeh, J. 2006. Real Analysis: Theory of Measure and Integration, 2nd ed. Singapore:

World Scientific.

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