Gaussian measures in Hilbert Spaces and their applications.
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
ABSTRAK
Boby Gunarso. 2016. Gaussian Measures In Hilbert Spaces And Their Applications.
Skripsi. Program Studi Matematika, Jurusan Matematika, Fakultas Sains dan Teknologi,
Universitas Sanata Dharma, Yogyakarta.
Ukuran Lebesgue memegang peranan penting di ℝ � . Ukuran Lebesgue dikarakterisasi
secara tunggal oleh sifat hingga lokal dan invarian terhadap translasi. Dapat dipertanyakan
apakah ukuran Lebesgue memiliki makna di ruang berdimensi takterhingga. Jawabannya adalah
tidak. Untuk mengkonstruksikan ukuran yang terdefinisi dengan baik (well defined) di dalam
ruang berdimensi tak terhingga, kita dapat menambahkan faktor eksponensial yang turun secara
cepat ke dalam ukuran Lebesgue sehingga diperoleh apa yang disebut ukuran Gaussian.
Sayangnya, ukuran Gaussian dengan operator identitas sebagai fungsi kovariansi masih tidak
dapat didefinisikan di ruang Hilbert separabel berdimensi tak terhingga. Terdapat setidaknya 2
cara untuk mengatasi masalah ini. Pertama, kita dapat menggunakan operator trace class sebagai
fungsi kovariansi untuk menunjukkan eksistensi dari ukuran Gaussian di dalam ruang Hilbert
separabel berdimensi tak terhingga. Cara kedua dapat dilakukan dengan mempertahankan
operator kovariansi identitas tetapi akibatnya ukuran tersebut hanya terdefinisi pada ruang dual
topologi dari ruang nuklir. Dalam skripsi ini, kita akan fokus pada pendekatan pertama, yaitu
mengkonstruksikan ukuran Gaussian di dalam ruang Hilbert separabel berdimensi takterhingga
dengan menggunakan operator trace class sebagai fungsi kovariansi. Kita akan mulai konstruksi
dari ukuran Gaussian pada garis real, pada ruang Euklidean berdimensi hingga, dan akhirnya
pada sebarang ruang Hilbert separabel berdimensi takterhingga. Kita juga akan menggunakan
ukuran Gaussian untuk mempelajari variabel random Gaussian, pemetaan derau putih, turunan
Malliavin, dan konstruksi dari gerak Brown di ruang Hilbert Gaussian.
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
ABSTRACT
Boby Gunarso. 2016. Gaussian Measures In Hilbert Spaces And Their Applications. A
Thesis. Mathematics Study Program, Department of Mathematics, Faculty of Science and
Technology, Sanata Dharma University, Yogyakarta.
The Lebesgue measure plays a fundamental role in ℝ � . 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 defined measure in infinite dimensional spaces, we
can incorporate a rapidly decreasing exponential factor to the Lebesgue measure and hence we
obtain the so-called Gaussian 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 dimensional
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 construct 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.
,
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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
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
i
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
BACHELOR TI{ESIS
GAUSSIAN MEASURES IN HILBERT SPACES AND THEIR
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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
BACHELOR THESIS
GAUSSIAN MEASURES IN HILBERT SPACES AND THEIR
APPLICATIONS
Written by:
Boby Gunarso
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Sudi Mungkasi, S.Si., M.Math.Sc., Ph.D.
iii
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 references, as a scientific paper should.
Yogyakarta, 1B July 2016
The Writer
[1l[,
Boby Gunarso
IV
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
ABSTRACT
Boby Gunarso. 2016. Gaussian Measures In Hilbert Spaces And Their Applications. 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 defined measure in infinite dimensional spaces, we can incorporate a rapidly decreasing
exponential factor to the Lebesgue measure and hence we obtain the so-called Gaussian 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 dimensional 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 construct 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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 Universitas Sanata Dharma, karya ilmiah saya yang berjudul:
GAUSSIAN MEASURES IN HILBERT SPACES AND THEIR
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
'\r"ry
,\l
Boby Gunarso
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 people. 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 support 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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
D.
Motivation to Measure in Infinite Dimensional Spaces . . . . . . . . .
CHAPTER 3 GAUSSIAN MEASURES IN HILBERT SPACES
41
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 D1,2 ♣H , µ q . . . . . . . . . . . . . . .
76
H.
Brownian Motion in Gaussian Hilbert Space . . . . . . . . . . . . . .
79
CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS
85
A.
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
B.
Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
BIBLIOGRAPHY
87
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
CHAPTER 1
INTRODUCTION
A.
Research Background
In Euclidean geometry we need the concept of a measure to know the measure µ ♣Sq 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 properties 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 R④Q form a partition of R
into disjoint cosets. Next, by the denseness of each coset A P R④Q, we can have the
following set of all coset representatives V
✏ txA : xA P A ❳r0, 1s and A P R④Q✉ called
Vitali set. Using the countable additivity and translation invariant properties of a measure, 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 separable Banach space is the trivial measure. So we will introduce the so-called Gaussian
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 derivative, 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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
E. Research Benefits
F. Research Methods
G. Systematics Writing
CHAPTER II. TOOLS FROM MEASURE THEORY AND FUNCTIONAL ANALYSIS
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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
G. The Malliavin-Sobolev Space D1,2 ♣H , µ q
H. Brownian Motion in Gaussian Hilbert Space
CHAPTER V. CONCLUSIONS AND RECOMMENDATIONS
A. Conclusions
B. Recommendations
BIBLIOGRAPHY
5
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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
❸ 2Ω a σ -
algebra on Ω if it satisfies the following axioms
(i) (Empty set)
❍ P F.
P
(ii) (Complement) If A F , then Ac
P F.
P
(iii) (Countable unions) If A1 , ..., An , ... F , then
♣
q
➈✽
n✏1 An
P F.
P
We also call the ordered pair Ω, F as a measurable space and A F as a measurable 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
t❍, Ω✉ 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
♣ q
❸ 2Ω.
We define σ G to be the intersection of all the σ -algebras on Ω that contain G and
6
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
we call it the σ -algebra generated by G . Clearly, σ ♣G 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 B ♣Ωq, as the Borel σ -algebra on Ω, i.e., B ♣Ωq ✏ σ ♣τ q where τ is a
topology on Ω.
From the definition above, we can directly show that if P♣Aq=”the set A has property P” and satisfies the following axioms
(i) P♣Aq is true for every A P G .
(ii) P♣❍q is true.
(iii) If P♣Aq is true for some A ❸ Ω, then P♣Ac q is also true.
(iv) If An ❸ Ω and P♣An q is true for every n P N, then P♣
➈
nPN An
q is also true,
then P♣Aq is true for every A P σ ♣G 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 A1 ❳ A2 P A for every A1 , A2 P A .
We call B a λ -system if it satisfies the following axioms
(i) (Empty set) ❍ P B.
(ii) (Complement) If B P B, then Bc P B.
(iii) (Countable disjoint unions) If B1 , ..., Bn , ... P B and if Bi ❳ B j ✏ ❍ for i ✘ j, then
➈✽
n✏1 Bn
P B.
Clearly a λ -system which is also a π -system is a σ -algebra since
➈✽
n✏1 Bn
✏ B1 ❨
♣B2③B1q ❨ ♣B3③B2③B1q ❨ ... P B for any sequence ♣BnqnPN in B. Moreover, every
λ -system B is closed under proper differences, i.e., if B1 , B2 P B with B2 ❸ B1 , then
B1 ③B2 P B. Now we will prove the following lemma before stating the Dynkin’s π ✁ λ
theorem.
7
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Lemma 2.1.1. Let A be a π -system of Ω and l ♣A q be the smallest λ -system containing A as a subset. Then, l ♣A q is a σ -algebra.
Proof. It suffices to prove that l ♣A q is a π -system. Set lA ✏ tB ❸ Ω : A ❳ B P l ♣A q✉
for any A P l ♣A q. It is clear that
❍ P lA and lA is closed under countable disjoint
unions. Next, if B P lA , then A ❳ Bc ✏ A③♣A ❳ Bq is a proper difference of sets in the
λ -system l ♣A q, and so is in lA . This shows that lA is a λ -system for any A P l ♣A q.
Now suppose A P A . Then A ❳ B P A ❸ l ♣A q for every B in the π -system A . Hence,
B P lA and thus lA ❹ A . But since l ♣A q is the smallest λ -system containing A , then
we have lA ❹ l ♣A q. Therefore, A ❳ B P l ♣A q for every B P l ♣A q. Finally, let B P l ♣A q
and consider lB ✏ tA ❸ Ω : A ❳ B P l ♣A q✉. Using our result before, we conclude that
lB is a λ -system and lB ❹ A . But this means A ❳ B P l ♣A q for every A P l ♣A q. Since
B P l ♣A 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
❸ B. Then, σ ♣A q ❸ B.
B a λ -system of subsets of Ω such that A
Proof. Let Ω be a nonempty set and A , B be π -system and λ -system of Ω, respec-
❸ B. Then, by definition of l ♣A q as the smallest λ -system containing A , we have l ♣A q ❸ B. From the previous lemma, we know that l ♣A q is a
σ -algebra, and since A ❸ l ♣A q, we have σ ♣A q ❸ l ♣A q. This completes the proof,
since l ♣A q ❸ B.
tively, such that A
Definition 2.1.4 (Measure). Let ♣Ω, F q be a measurable space. If µ : F
R is the extended nonnegative real number line R
ÑR
, where
✏ r0, ✽s, satisfies the following
axioms
(i) (Null empty set) µ ♣❍q ✏ 0
(ii) (Countable additivity) If A1 , ..., An , ... is a sequence of pairwise disjoint sets in
➈
F , then µ ♣ ✽
n✏1 An q ✏
➦ ✽ µ ♣A q,
n
n✏1
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
then we call µ a measure on ♣Ω, F q and ♣Ω, F , µ q a measure space. If µ ♣Ωq ✏ 1,
then we call µ a probability measure and ♣Ω, F , µ q a probability space.
Definition 2.1.5 (Outer Measure). Let Ω be an arbitrary set. A set function µ ✝ : 2Ω Ñ
R is called an outer measure on Ω if it satisfies the following axioms
(i) (Null empty set) µ ✝ ♣❍q ✏ 0.
(ii) (Monotonicity) If A1 , A2 P 2Ω such that A1 ❸ A2 , then µ ✝ ♣A1 q ↕ µ ✝ ♣A2 q.
➈
(iii) (Countable subadditivity) If A1 , ..., An , ... P 2Ω , then µ ✝ ♣ ✽
n✏1 An q ↕
➦✽
n✏1 µ
✝ ♣An q.
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., µ ✝ ♣A1 ❨ A2 q ✏ µ ✝ ♣A1 q
➈
µ ✝ ♣A2 q for every A1 , A2 P
q ✏ ➦Nn✏1 µ ✝♣Anq
➈ A q➙
for every N P N and pairwise disjoint sets A1 , ..., An , ... P 2Ω so that µ ✝ ♣ ✽
n✏1 n
➦✽ µ ✝♣A q. Using this result together with the countable subadditivity of µ ✝, we
n
n✏1
✝
2Ω such that A1 ❳ A2 ✏ ❍, then we have µ ✝ ♣ ✽
n✏1 An q ➙ µ ♣
➈N
n✏1 An
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
µ ✝ ♣ T q ✏ µ ✝ ♣ T ❳ Aq
µ ✝ ♣T ❳ Ac q for any set T
P 2Ω .
Definition 2.1.6 (Covering class of a set). Let Ω be an arbitrary set and A
❸ 2Ω be a
collection of subsets of Ω such that
(i) (Empty set) ❍ P A .
(ii) (Countable covering) Ω has a countable covering in A , i.e., there exist a sequence A1 , ..., An , ... P A such that
➈✽
n✏1 An
then we call A as the covering class of Ω
9
✏ Ω,
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 , ... P A such that
➈✽
n✏1 An
❹ 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
µ ✝ : 2Ω Ñ r0,
Ñ r0, ✽s such that δ ♣❍q ✏ 0. Then, the set function
✽s defined by
✰
★➳
✽
✽
↕
µ ✝ ♣Aq ✏ inf
δ ♣An q : A1 , ..., An , ... P A and
An ❹ A
n✏1
n✏1
is an outer measure on Ω based on δ .
Proof. It is obvious that µ ✝ ♣Aq P r0,
and hence µ ✝ ♣❍q ✏ 0.
✽s for every A P 2Ω and µ ✝♣❍q ↕ δ ♣❍q ✏ 0
Next, if A1 , A2 P 2Ω such that A1 ❸ 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 µ ✝ ♣A1 q and
µ ✝ ♣A2 q, respectively. Then, M ❸ N and consequently µ ✝ ♣A1 q ✏ inf M ↕ inf N ✏ µ ✝ ♣A2 q
which proves the monotonicity of µ ✝ . Last, let A1 , ..., An
→ 0, then
➈ A ❹ A and
for each n P N there exist a sequence ♣An,k qkPN in A such that ✽
n
k✏1 n,k
➦✽ δ ♣A q ↕ µ ✝♣A q ε . Therefore, ➈✽ ♣➈✽ A q ❹ ➈✽ A which implies
n
n,k
k✏1
n✏1
k✏1 n,k
n✏1 n
2
P 2Ω.
Given ε
n
that
✽
✽
✽
✽ ➳
✽
➳
↕
➳
➳
ε
✝
✝
µ ♣ An q ↕
r δ ♣An,k qs ↕ ♣µ ♣Anq 2n q ✏ µ ✝♣Anq ε
n✏1
n✏1
n✏1 k✏1
n✏1
and since ε
→ 0 is arbitrary, the conclusion follows.
Definition 2.1.7 (Lebesgue measure). Let I0 be the collection of ❍ and all open intervals in R. Let δ be a set function on I0 , i.e., δ : I0 Ñ r0,
✽s such that δ ♣❍q ✏ 0 and
➈ I q ✏ ➦ ✽ δ ♣I q.
for any sequence ♣In qnPN of disjoint intervals in R, we have δ ♣ ✽
n
n✏1 n
n✏1
If I is an interval in R with endpoints a, b P R where a ➔ b, then we define δ ♣I q ✏ b ✁ a
and if I is an infinite interval in R, we define δ ♣I q ✏ ✽. Define the Lebesgue outer
10
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
measure on R, µ ✝ : 2R Ñ r0,
µ ✝ ♣Aq ✏ inf
★✽
➳
n✏1
✽s, as follows
δ ♣An q : A1 , ..., An , ... P I0 and
✽
↕
n✏1
✰
An ❹ A
We call the restriction of µ ✝ to the collection of all Lebesgue measurable (i.e. measurable 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 µ ✝ ♣Aq ✏ 0.
Proof. First we will show that every singleton has measure zero. Let x P R and ε
→
0. Then ♣x ✁ ε2 , x ε2 q is an open cover of tx✉. Hence, we have µ ✝ ♣tx✉q ↕ δ ♣♣x ✁
ε
ε
✝
2,x
2 qq ✏ ε . Since ε → 0 is arbitrary, it follows that µ ♣tx✉q ✏ 0. Now let A be a
➈
countable subset of R, then A ✏ nPN txn ✉ is a countable union of null sets An ✏ txn ✉
➦
and therefore µ ✝ ♣Aq ↕ nPN µ ✝ ♣An q ✏ 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 ⑨
♣a ✁ ε2 , b ε2 q P I0, i.e., ♣a ✁ ε2 , b ε2 q is an open cover in I0 of I. Hence, we have
µ ✝ ♣I q ↕ δ ♣♣a ✁ ε2 , b ε2 qq ✏ δ ♣I q ε . Since ε is arbitrary, then µ ✝ ♣I q ↕ δ ♣I q. On the
➦
other side, let ♣In qnPN ⑨ I0 be a covering sequence of I. We will show that nPN δ ♣In q ➙
➦
δ ♣I q. If there exist k such that Ik P ♣In qnPN is an infinite interval, then nPN δ ♣In q ➙
δ ♣Ik q ✏ ✽ → δ ♣I q and we are done. So we assume that each In is a finite open
interval. Let ♣Jn qnPN be a subsequence of ♣In qnPN such that Jn ❳ I ✘ ❍ for every n and
➦
➦
Jk ❶ Jl if k ✘ l. Then nPN δ ♣Jn q ↕ nPN δ ♣In q. Since I is compact and ♣Jn qnPN is
an open cover of I, then there exist a finite subcover ♣Jn qN
k✏1 ⑨ ♣Jn qnPN of I. Assume
that Jn ✏ ♣an , bn q with an ↕ ... ↕ an . Obviously, if an ✏ an for i ✘ j, then we have
Jn ⑨ Jn or vice versa which contradicts the fact that Jk ❶ Jl if k ✘ l. Therefore, we
have an ➔ ... ➔ an . Next we will show that am 1 ➔ bm for every m P tn1 , ..., nN ✁1 ✉.
k
i
i
i
i
1
N
i
j
1
N
11
j
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
➙ bm for some m. Since Jn ❳ I ✘ ❍ for every n, then there exist
xm P Jm ❳ I and xm 1 P Jm 1 ❳ I such that am ➔ xm ➔ bm ↕ am 1 ➔ xm 1 ➔ bm 1 .
Since I is an interval and xm , xm 1 P I, we have rxm , xm 1 s ⑨ I. Consider the following
➈ J . If b ➔ a , then
two possibilities. If bm ✏ am 1 , then bm P I but bm ❘ N
m
m 1
i✏1 n
rbm, am 1s ⑨ I but rbm, am 1s ❶ ➈Ni✏1 Jn which contradicts the fact that ♣Jn qNi✏1 is an
open cover of I. Thus we have am 1 ➔ bm for every m P tn1 , ..., nN ✉. So we have the
If not, then am
1
i
i
i
following inequalities
➳
nPN
δ ♣In q ➙
➳
nPN
δ ♣Jn q ➙
➳N
i✏1
δ ♣Jni q ✏ ♣bn1 ✁ an1 q
➙ ♣ an ✁ an q
➙ bn ✁ an
➙ b✁a
✏ δ ♣I q,
2
1
N
i.e., δ ♣I q is a lower bound of
➦
nPN δ
...
♣bn ✁ ✁ an ✁ q ♣bn ✁ an q
...
♣an ✁ an ✁ q ♣bn ✁ an q
N 1
N
N
N 1
N 1
N
N
N
1
♣Inq so that µ ✝♣I q ➙ δ ♣I q and the conclusion
follows. Second, let I ✏ ♣a, 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
µ ✝ ♣♣a, bqq ↕ µ ✝ ♣ra, bsq ↕ µ ✝ ♣ta✉q
µ ✝ ♣♣a, bqq
µ ✝ ♣tb✉q ✏ µ ✝ ♣♣a, bqq
which implies that µ ✝ ♣♣a, bqq ✏ µ ✝ ♣ra, bsq ✏ δ ♣ra, bsq ✏ δ ♣♣a, bqq. Third, if I is a finite
✏ ♣a, bs, then we have µ ✝♣♣a, bqq ↕ µ ✝♣♣a, bsq ↕ µ ✝♣♣a, bqq
µ ✝ ♣tb✉q ✏ µ ✝ ♣♣a, bqq so that µ ✝ ♣♣a, bsq ✏ µ ✝ ♣♣a, bqq ✏ δ ♣♣a, bqq ✏ δ ♣♣a, bsq. Clearly
for I ✏ ra, bq we also have µ ✝ ♣ra, bqq ✏ δ ♣ra, bqq. Last, if I is an infinite interval, say
of the form I ✏ ♣a, ✽q, then we have µ ✝ ♣♣a, ✽qq ➙ µ ✝ ♣♣a, nqq ✏ δ ♣♣a, nqq ✏ n ✁ a
for any n → a so that µ ✝ ♣♣a, ✽qq ✏ ✽ ✏ δ ♣♣a, ✽qq. Similar arguments also hold
interval of the form I
12
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
for any other infinite interval I.
Definition 2.1.8 (Translation Invariance). Let Ω be a vector space, ♣Ω, F , µ q a measure space, and A
x ✏ ta
x : a P A✉ for any A P F 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 P Ω, A x has the same measure
as A itself.
(iii) ♣Ω, F , µ q 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 ♣In qnPN of A in I0 , then we have
➈
xq ✏ ♣ ✽
n✏1 In q
x ❹ A and hence
➈ ✽ ♣I
n✏1 n
➦ ✽ δ ♣I q ✏ ➦ ✽ δ ♣I x q ➙ µ ✝ ♣A x q.
n
n
n✏1
n✏1
Since
♣InqnPN is an arbitrary open interval cover of A, then µ ✝♣Aq ➙ µ ✝♣A xq from the
➦ δ ♣I q. Conversely, by applying the similiar
definition of µ ✝ as the infimum of ✽
n
n✏1
method to the set A x and its translate ♣A xq✁ x ✏ A, we have the reverse inequality
µ ✝ ♣A xq ➙ µ ✝ ♣Aq 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
µ ✝ ♣T ❳♣A
xqq
µ ✝ ♣T ❳♣A
xqc q ✏ µ ✝ ♣tT ❳♣A
13
xq✉✁ xq
µ ✝ ♣tT ❳♣A
xqc ✉✁ xq
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
✏ µ ✝♣♣T ✁ xq❳ Aq
✏ µ ✝ ♣T ✁ x q
✏ µ ✝ ♣T q
µ ✝ ♣♣T ✁ xq❳ Ac 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
µ ♣A
x q ✏ µ ✝ ♣A
xq ✏ µ ✝ ♣Aq ✏ µ ♣Aq,
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 ♣Ω, F , µ q 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 ♣Ω, F0 , µ0 q such that F0
❹F
and µ0
✏µ
on F , then we say that ♣Ω, F0 , µ0 q is a complete extension of ♣Ω, F , µ q. If it is the
smallest complete extension of ♣Ω, F , µ q, i.e, if for any complete extension ♣Ω, F1 , µ1 q
of ♣Ω, F , µ q we have F1 ❹ 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
P 2R, we have 0 ↕ µ ✝♣T ❳ Aq ↕ µ ✝♣Aq ✏ 0
and hence µ ✝ ♣T ❳ Aq ✏ 0. Therefore, µ ✝ ♣T q ↕ µ ✝ ♣T ❳ Ac q ✏ µ ✝ ♣T ❳ Aq µ ✝ ♣T ❳ Ac q
measurable. Clearly, for any testing set T
so that A is Lebesgue measurable. Next, assume that A has Lebesgue measure zero and
B ⑨ A, then µ ✝ ♣Bq ↕ µ ✝ ♣Aq ✏ µ ♣Aq ✏ 0 so that B is Lebesgue measurable.
Since every interval is Lebesgue measurable and every open set is a union of count14
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 ♣X, A q and ♣Y, B q be measurable spaces.
A function f : X
Ñ Y is called measurable (B ✁ A -measurable) if for all B P B, we
have f ✁1 ♣Bq P A .
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 ♣X, A q, ♣Y, B q be measurable spaces and G
❸ 2Y be the generator
of B, i.e., B ✏ σ ♣G q. Then a function f : X Ñ Y is measurable if and only if f ✁1 ♣Gq P
A for every G P G .
Proof. Clearly if f is measurable, then f ✁1 ♣Gq P A for every G P G . Define M
✥
✭
✏
M P B : f ✁1 ♣M q P A . It is easy to show that M is a σ -algebra as follows: since
f ✁1 ♣Y q ✏ X
P A , then Y P M . If M P M , then f ✁1♣Mq P A , and hence ♣ f ✁1♣Mqqc ✏
f ✁1 ♣M c q P A so that M c P M . Finally, if M1 , ..., Mn , ... P M , then f ✁1 ♣Mi q P A for
➈
➈
every i P N which implies that ni✏1 f ✁1 ♣Mi q ✏ f ✁1 ♣ ni✏1 ♣Mi qq P A . Therefore, since
G ❸ M , then σ ♣G q ❸ σ ♣M q, i.e, B ❸ 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
ak
P R and Ek ✏ φ ✁1♣tak ✉q.
✏ ➦nk✏1 ak 1E
k
where
This definition of simple functions play a fundamental
role in measure theory. This is also an implication of the following theorem.
15
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Theorem 2.1.7 (Approximation by simple functions). If f : X
ÑR
is a nonnegative
measurable function, then there is a monotone increasing sequence of nonnegative
simple functions ϕn : X
ÑR
such that ϕn Ñ f pointwise as n Ñ
✽.
Proof. For each n P N, we divide the interval r0, 2n q in the range of f into 22n subinter-
vals of width 2✁n as follows. Define Ik,n ✏ rk2✁n , ♣k
1q2✁n q for k ✏ 0, 1, ..., 22n ✁ 1
✽s. We also define Ek,n ✏ f ✁1♣Ik,nq and En ✏ f ✁1♣Inq. Then, it is easy to
➦
see that the sequence of simple functions ♣ϕn qnPN where ϕn ✏ k2✏✁01 k2✁n 1E
2n 1 E
and In ✏ r2n ,
n
k,n
n
satisfies the required properties.
Note that every function f can be written as a sum of two positive functions,
✏ f ✁ f ✁ where f ✏ max t f , 0✉ and called the positive part of f and f ✁ ✏
max t✁ f , 0✉ and called the negative part of f . This means that we can prove that some
i.e., f
properties hold for f just simply by proving that it is true for the corresponding simple
functions.
Definition 2.1.12 (Lebesgue Integral). Let ♣Ω, F , µ q be a measure space. The Lebesgue
integral over Ω of a measurable simple function φ : Ω Ñ R is defined as
➺
φ dµ
Ω
✏
➺ ➳
n
Ω k✏1
ak 1Ek d µ
✏
n
➳
k✏1
ak µ ♣Ek q.
If f : Ω Ñ R is a nonnegative measurable function, then the Lebesgue integral of f
➩
✥➩
✭
✏ sup Ω φ d µ : φ is simple and 0 ↕ φ ↕ f . Finally, for
Ω
any measurable function f : Ω Ñ R ✏ R ❨t✁✽, ✽✉, the Lebesgue integral of f over
➩
➩
➩
Ω is defined as Ω f d µ ✏ Ω f d µ ✁ Ω f ✁ d µ .
over Ω is defined as
f dµ
Theorem 2.1.8 (Monotone Convergence Theorem). Let ♣ fn qnPN be a sequence of non-
negative measurable functions in a measure space ♣Ω, F , µ q which increasing pointwise to f , then
➺
f dµ
Ω
✏
➺
lim fn d µ
Ω nÑ
✽
16
✏ nÑlim✽
➺
fn d µ .
Ω
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Proof. Since fn is measurable and increasing pointwise to f , then f is measurable.
➩
➩
↕ f , then Ω fnd µ ↕ Ω f d µ . Take any c P ♣0, 1q and fixed a simple
function 0 ↕ ϕ ↕ f . Let An ✏ tx P Ω : fn ♣xq ➙ cϕ ♣xq✉. It is clear that An is increasing.
➈
To show that X ✏ nPN An , let x P Ω. Then
Also, since fn
(i) If ϕ ♣xq ✏ 0, then x P An for any n P N.
(ii) If ϕ ♣xq → 0, from the fact that f ♣xq ➙ ϕ ♣xq → cϕ ♣xq, then there exist n P N such
that fn ♣xq ➙ t ϕ ♣xq.
This means that x P An . For any A P F , set v♣Aq ✏
where ϕ
✏ ➦mn✏1 yn1E
n
➩
✏ c ➦mn✏1 yn µ ♣A ❳ Enq
A cϕ d µ
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
➺
cϕ d µ
Ω
✏ v♣ Ωq ✏ v♣
↕
nPN
An q ✏ lim v♣An q ✏ lim
nÑ
✽
nÑ
✽
↕ nÑlim✽
↕ nÑlim✽
That is, c
limnÑ ✽
➩
Ω ϕdµ
➩
↕ limnÑ
Ω fn d µ .
✽
➩
Ω fn d µ .
➺
cϕ d µ
➺
An
➺
An
fn d µ
fn d µ .
Ω
Since c P ♣0, 1q is arbitrary, we have
➩
Ω ϕdµ
↕
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 ♣Ω, F , µ q
be a measure space and f : A③N Ñ R be a measurable function on A③N where A, N P
F , A ❹ N, and N is a null set. We write
➩
A
17
f d µ for
➩
A
frd µ where fr is a nonnegative
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
extended real valued measurable function defined on A by setting
✩
✬
✫
rf ♣xq ✏ f ♣xq
✬
✪0
if x P A③N
if x P N
.
➩ frd µ ✏ ➩ frd µ ➩ frd µ ✏ ➩ frd µ .
A
A③N
N
A③N
➩
The definition above of A f d µ for a function f which is defined only a.e. (almost
➩ f dµ.
everywhere) on A is for convenience of not having to write
It is a fact that
A③N
Theorem 2.1.9 (Fatou’s lemma). Let ♣Ω, F , µ q be a measure space and ♣ fn qnPN be an
arbitrary sequence of nonnegative extended real valued measurable functions on a set
A P F , then we have
➺
lim inf fn d µ
A nÑ
In particular, if f
✏ limnÑ
✽
↕ lim
inf
nÑ ✽
➺
fn d µ .
A
✽ fn exist a.e. on A, then
➺
f dµ
A
Proof. By definition, lim infnÑ ✽ fn
↕ lim
inf
nÑ ✽
➺
✏ limnÑ
fn d µ .
A
✽ infk➙n fk . Clearly, ♣infk➙n fk qnPN is
increasing and hence by the monotone convergence theorem
➺
➺
➺
lim inf fk d µ ✏ lim
inf fk d µ
lim inf fn d µ ✏
nÑ ✽ A k➙n
A nÑ ✽ k➙n
A nÑ ✽
➺
✏ lim
inf inf fk d µ
nÑ ✽ A k➙n
➺
↕ lim
inf
nÑ ✽
Last, if f
✏ limnÑ
✽ fn exist a.e. on A, then f
✏ lim infnÑ
fn d µ
A
✽ fn a.e. on A. Thus the
conclusion follows by Definition 2.12.
Theorem 2.1.10 (Lebesgue dominated convergence theorem). Let ♣Ω, F , µ q be a mea18
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Ñ R be a sequence of measurable functions such that fn Ñ f
pointwise as n Ñ ✽. Suppose that there exist an integrable function g such that ⑤ fn ⑤ ↕
➩
g for every n P N. Then fn and f are also integrable and limnÑ ✽ Ω ⑤ fn ✁ f ⑤d µ ✏ 0.
sure space and fn : X
Proof. Obviously fn and f are integrable. Also, 2g ✁⑤ fn ✁ f ⑤ is nonnegative and measurable. By Fatou’s lemma
➺
lim inf♣2g ✁⑤ fn ✁ f ⑤qd µ
Ω nÑ
✽
➺
↕ lim
inf ♣2g ✁⑤ fn ✁ f ⑤qd µ
nÑ ✽
Ω
Since fn Ñ f , the left hand side of the inequality above is just
➩
Ω 2gd µ
and the right
hand side is equal to
✂➺
lim inf
nÑ
✽
Ω
2gd µ ✁
✡
➺
Ω
⑤ f n ✁ f ⑤d µ ✏
✏
➺
Ω
Ω
2gd µ ✁ lim inf
nÑ
✂➺
✽
2gd µ ✁ lim sup
nÑ
Ω
✂➺
✽
⑤ f n ✁ f ⑤d µ
Ω
✡
⑤ f n ✁ f ⑤d µ
✡
➔ ✽, we may cancel the above inequality to obtain
✟
➩
➩
Ω ⑤ f n ✁ f ⑤d µ ↕ 0 and so limnÑ ✽ Ω ⑤ f n ✁ f ⑤d µ ✏ 0.
Using the fact that
lim supnÑ ✽
➩
➺
Ω 2gd µ
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 ⑤ fn ⑤ ↕ g a.e.
(ii) By the triangle inequality, it is clear that limnÑ ✽
➩
Ω
fn d µ
✏
➩
Ω
f dµ.
Theorem 2.1.11 (Beppo-Levi). Let ♣Ω, F , µ q be a measure space and fn : Ω Ñ r0,
be a sequence of nonnegative measurable functions. Then
➺ ➳
Ω nPN
fn d µ
✏
➳➺
nPN Ω
19
fn d µ .
✽s
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Proof. Let gN
✏ ➦Nn✏1 fn and g ✏ ➦nPN fn.
By the monotone convergence theorem,
we have
➺
gd µ
Ω
✏
➺
lim gN d µ
Ω NÑ
✽
✏ NÑlim✽
➺
✏ NÑlim✽
gN d µ
Ω
➳N ➺
n✏1 Ω
fn d µ
✏
✽➺
➳
n✏1 Ω
fn d µ .
B. Tools from Functional Analysis
Definition 2.2.1 (Normed space). A normed space is a vector space X over a field
P tR, C✉ equipped with a function ⑥ ☎ ⑥ : X Ñ R, called a norm, such that for every
x, y P X and λ P F the following axioms hold
F
(i) (Nullity) ⑥x⑥ ✏ 0 implies x ✏ 0.
(ii) (Homogeinity) ⑥λ x⑥ ✏ ⑤λ ⑤⑥x⑥.
(iii) (Triangle inequality) ⑥x
y⑥ ↕ ⑥ x ⑥
⑥ y⑥ .
From these axioms, we can easily deduce that ⑥x⑥ ➙ 0 and ⑥x⑥ ✏ 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) d ♣x, yq ✏ 0 if and only if x ✏ y.
(ii) (Symmetry) d ♣x, yq ✏ d ♣y, xq.
(iii) (Triangle inequality) d ♣x, yq ↕ d ♣x, zq
d ♣z, yq.
From the third axiom, we also can easily deduce that d ♣x, yq ➙ 0 for every x, y P X. Also
notice that every normed space is metrizable, i.e., we can generate a metric d ♣x, yq :✏
⑥x ✁ y⑥ for every x, y P X in a normed space.
20
This is the standard metric that will be
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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
♣xnqnPN 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 d ♣xn , xq ➔ ε for every n ➙ N. A sequence ♣xn qnPN in X is called a
Cauchy sequence if for every ε → 0, there exist N P N such that d ♣xn , xm q ➔ 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 ✏ n1 , n P N in the metric space of real interval ♣0, 1q with the
usual metric d ♣x, yq ✏ ⑤x ✁ y⑤ which is Cauchy but not convergent to an element in
♣0, 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, z P X and λ
①☎, ☎② : X ✂ X Ñ F that satisfies the following axioms for any
P F.
(i) (Conjugate Symmetry) ①x, y② ✏ ①y, x②.
(ii) (Distributive) ①x
y, z② ✏ ①x, z②
①y, z②.
(iii) (Homogeneity) ①λ x, y② ✏ λ ①x, y②.
(iv) (Positive definiteness) ①x, x② ➙ 0 and ①x, x② ✏ 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., ⑥x⑥ ✏ ①x, x② 2 for some inner product ①☎, ☎②
1
defined on the normed space H , we call H a Hilbert space.
21
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Definition 2.2.6 (Orthonormal Basis of Hilbert Spaces). Let ♣ei qiPI be a collection of
vectors in H . We call ♣ei qiPI an orthonormal basis of H if ①ei , e j ② ✏ 0 for i ✘ j and
①h, ei② ✏ 0 for every i P I implies that h ✏ 0.
Denote by Cb ♣H q the space of all continuous bounded mappings from H into R
endowed by the norm ⑥ f ⑥0 ✏ supxPH ⑤ f ♣xq⑤ for any f
P Cb♣H q.
Proposition 2.2.1. Cb ♣H q is a Banach space.
Proof. Let ♣ fn qnPN be a Cauchy sequence in Cb ♣H q. Since ⑤ fn ♣xq ✁ fm ♣xq⑤ ↕ ⑥ fn ✁
fm ⑥0 , the sequence ♣ fn ♣xqqnPN 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 ♣xq ✏ limnÑ ✽ fn ♣xq is an R-
valued function. Now we will show that the limit f is a bounded function as fol-
→ 0 and N be a positive integer such that ⑥ fn ✁ fm⑥0 ➔ ε for all n, m ➙ N.
Then, the inequality ⑤ f ♣xq⑤ ↕ ⑤ f ♣xq ✁ fN ♣xq⑤ ⑤ fN ♣xq⑤ ↕ ⑥ f ✁ fN ⑥0 ⑥ fN ⑥0 holds for
all x P H . But since for n ➙ N, ⑤ fn ♣xq✁ fN ♣xq⑤ ➔ ε for every x, then ⑤ f ♣xq✁ fN ♣xq⑤ ✏
limnÑ ✽ ⑤ fn ♣xq ✁ fN ♣xq⑤ ↕ ε so that ⑥ f ✁ fN ⑥0 ✏ supxPH ⑤ f ♣xq ✁ fN ♣xq⑤ ↕ ε . This
lows: Let ε
shows that the function f is bounded. Let x0 in H . By the continuity of fN , choose
→ 0 such that ⑤ fN ♣xq ✁ fN ♣x0q⑤ ➔ ε whenever ⑥x ✁ x0⑥ ➔ δ . Then if ⑥x ✁ x0⑥ ➔ δ ,
we have ⑤ f ♣xq ✁ f ♣x0 q⑤ ↕ ⑤ f ♣xq ✁ fN ♣xq⑤ ⑤ fN ♣xq ✁ fN ♣x0 q⑤ ⑤ fN ♣x0 q ✁ f ♣x0 q⑤ ↕ ⑥ f ✁
fN ⑥0 ⑤ fN ♣xq ✁ fN ♣x0 q⑤ ⑥ fN ✁ f ⑥0 ➔ ε ε ε ✏ 3ε . This shows the continuity of
f . To finish the proof we need to show fn converges in norm, i.e. ⑥ f ✁ fn ⑥0 Ñ 0
as n Ñ ✽. This is clear by using the triangle inequality as follows: ⑥ f ✁ fn ⑥0 ↕
⑥ f ✁ fN ⑥0 ⑥ fN ✁ fn⑥0 ↕ ε ε ✏ 2ε so that ⑥ f ✁ fn⑥0 Ñ 0.
δ
Theorem 2.2.1 (The orthogonal decomposition theorem). Let H be a Hilbert space
and S ❸ H a closed subspace of H . Then the orthogonal complement S❑ defined by
S❑ ✏ tx P X : ①x, y② ✏ 0 for every y
P S✉ is also a closed subspace of H
be represented as the direct sum of S and S❑ , i.e., H ✏ S ❵ S❑ .
22
and H can
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Proof. It is clear that S❑ is a vector sub
ABSTRAK
Boby Gunarso. 2016. Gaussian Measures In Hilbert Spaces And Their Applications.
Skripsi. Program Studi Matematika, Jurusan Matematika, Fakultas Sains dan Teknologi,
Universitas Sanata Dharma, Yogyakarta.
Ukuran Lebesgue memegang peranan penting di ℝ � . Ukuran Lebesgue dikarakterisasi
secara tunggal oleh sifat hingga lokal dan invarian terhadap translasi. Dapat dipertanyakan
apakah ukuran Lebesgue memiliki makna di ruang berdimensi takterhingga. Jawabannya adalah
tidak. Untuk mengkonstruksikan ukuran yang terdefinisi dengan baik (well defined) di dalam
ruang berdimensi tak terhingga, kita dapat menambahkan faktor eksponensial yang turun secara
cepat ke dalam ukuran Lebesgue sehingga diperoleh apa yang disebut ukuran Gaussian.
Sayangnya, ukuran Gaussian dengan operator identitas sebagai fungsi kovariansi masih tidak
dapat didefinisikan di ruang Hilbert separabel berdimensi tak terhingga. Terdapat setidaknya 2
cara untuk mengatasi masalah ini. Pertama, kita dapat menggunakan operator trace class sebagai
fungsi kovariansi untuk menunjukkan eksistensi dari ukuran Gaussian di dalam ruang Hilbert
separabel berdimensi tak terhingga. Cara kedua dapat dilakukan dengan mempertahankan
operator kovariansi identitas tetapi akibatnya ukuran tersebut hanya terdefinisi pada ruang dual
topologi dari ruang nuklir. Dalam skripsi ini, kita akan fokus pada pendekatan pertama, yaitu
mengkonstruksikan ukuran Gaussian di dalam ruang Hilbert separabel berdimensi takterhingga
dengan menggunakan operator trace class sebagai fungsi kovariansi. Kita akan mulai konstruksi
dari ukuran Gaussian pada garis real, pada ruang Euklidean berdimensi hingga, dan akhirnya
pada sebarang ruang Hilbert separabel berdimensi takterhingga. Kita juga akan menggunakan
ukuran Gaussian untuk mempelajari variabel random Gaussian, pemetaan derau putih, turunan
Malliavin, dan konstruksi dari gerak Brown di ruang Hilbert Gaussian.
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
ABSTRACT
Boby Gunarso. 2016. Gaussian Measures In Hilbert Spaces And Their Applications. A
Thesis. Mathematics Study Program, Department of Mathematics, Faculty of Science and
Technology, Sanata Dharma University, Yogyakarta.
The Lebesgue measure plays a fundamental role in ℝ � . 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 defined measure in infinite dimensional spaces, we
can incorporate a rapidly decreasing exponential factor to the Lebesgue measure and hence we
obtain the so-called Gaussian 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 dimensional
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 construct 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.
,
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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
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
i
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
BACHELOR TI{ESIS
GAUSSIAN MEASURES IN HILBERT SPACES AND THEIR
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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
BACHELOR THESIS
GAUSSIAN MEASURES IN HILBERT SPACES AND THEIR
APPLICATIONS
Written by:
Boby Gunarso
l23rr40a2
fl.#ffitr-#
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tr
ffi^ts
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Herry prflawanto
t"qS$J,
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Yogyakarta, 29 August 2016
Faculty of Science and Technology
,6("*;
ilffi
S
anata Dhanna
ljniversitv
Dean.
/*/^-
Sudi Mungkasi, S.Si., M.Math.Sc., Ph.D.
iii
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 references, as a scientific paper should.
Yogyakarta, 1B July 2016
The Writer
[1l[,
Boby Gunarso
IV
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
ABSTRACT
Boby Gunarso. 2016. Gaussian Measures In Hilbert Spaces And Their Applications. 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 defined measure in infinite dimensional spaces, we can incorporate a rapidly decreasing
exponential factor to the Lebesgue measure and hence we obtain the so-called Gaussian 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 dimensional 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 construct 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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 Universitas Sanata Dharma, karya ilmiah saya yang berjudul:
GAUSSIAN MEASURES IN HILBERT SPACES AND THEIR
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
'\r"ry
,\l
Boby Gunarso
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 people. 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 support 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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
D.
Motivation to Measure in Infinite Dimensional Spaces . . . . . . . . .
CHAPTER 3 GAUSSIAN MEASURES IN HILBERT SPACES
41
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 D1,2 ♣H , µ q . . . . . . . . . . . . . . .
76
H.
Brownian Motion in Gaussian Hilbert Space . . . . . . . . . . . . . .
79
CHAPTER 5 CONCLUSIONS AND RECOMMENDATIONS
85
A.
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
B.
Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
BIBLIOGRAPHY
87
ix
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
CHAPTER 1
INTRODUCTION
A.
Research Background
In Euclidean geometry we need the concept of a measure to know the measure µ ♣Sq 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 properties 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 R④Q form a partition of R
into disjoint cosets. Next, by the denseness of each coset A P R④Q, we can have the
following set of all coset representatives V
✏ txA : xA P A ❳r0, 1s and A P R④Q✉ called
Vitali set. Using the countable additivity and translation invariant properties of a measure, 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 separable Banach space is the trivial measure. So we will introduce the so-called Gaussian
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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.
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 derivative, 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
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
E. Research Benefits
F. Research Methods
G. Systematics Writing
CHAPTER II. TOOLS FROM MEASURE THEORY AND FUNCTIONAL ANALYSIS
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
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
G. The Malliavin-Sobolev Space D1,2 ♣H , µ q
H. Brownian Motion in Gaussian Hilbert Space
CHAPTER V. CONCLUSIONS AND RECOMMENDATIONS
A. Conclusions
B. Recommendations
BIBLIOGRAPHY
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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
❸ 2Ω a σ -
algebra on Ω if it satisfies the following axioms
(i) (Empty set)
❍ P F.
P
(ii) (Complement) If A F , then Ac
P F.
P
(iii) (Countable unions) If A1 , ..., An , ... F , then
♣
q
➈✽
n✏1 An
P F.
P
We also call the ordered pair Ω, F as a measurable space and A F as a measurable 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
t❍, Ω✉ 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
♣ q
❸ 2Ω.
We define σ G to be the intersection of all the σ -algebras on Ω that contain G and
6
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
we call it the σ -algebra generated by G . Clearly, σ ♣G 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 B ♣Ωq, as the Borel σ -algebra on Ω, i.e., B ♣Ωq ✏ σ ♣τ q where τ is a
topology on Ω.
From the definition above, we can directly show that if P♣Aq=”the set A has property P” and satisfies the following axioms
(i) P♣Aq is true for every A P G .
(ii) P♣❍q is true.
(iii) If P♣Aq is true for some A ❸ Ω, then P♣Ac q is also true.
(iv) If An ❸ Ω and P♣An q is true for every n P N, then P♣
➈
nPN An
q is also true,
then P♣Aq is true for every A P σ ♣G 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 A1 ❳ A2 P A for every A1 , A2 P A .
We call B a λ -system if it satisfies the following axioms
(i) (Empty set) ❍ P B.
(ii) (Complement) If B P B, then Bc P B.
(iii) (Countable disjoint unions) If B1 , ..., Bn , ... P B and if Bi ❳ B j ✏ ❍ for i ✘ j, then
➈✽
n✏1 Bn
P B.
Clearly a λ -system which is also a π -system is a σ -algebra since
➈✽
n✏1 Bn
✏ B1 ❨
♣B2③B1q ❨ ♣B3③B2③B1q ❨ ... P B for any sequence ♣BnqnPN in B. Moreover, every
λ -system B is closed under proper differences, i.e., if B1 , B2 P B with B2 ❸ B1 , then
B1 ③B2 P B. Now we will prove the following lemma before stating the Dynkin’s π ✁ λ
theorem.
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Lemma 2.1.1. Let A be a π -system of Ω and l ♣A q be the smallest λ -system containing A as a subset. Then, l ♣A q is a σ -algebra.
Proof. It suffices to prove that l ♣A q is a π -system. Set lA ✏ tB ❸ Ω : A ❳ B P l ♣A q✉
for any A P l ♣A q. It is clear that
❍ P lA and lA is closed under countable disjoint
unions. Next, if B P lA , then A ❳ Bc ✏ A③♣A ❳ Bq is a proper difference of sets in the
λ -system l ♣A q, and so is in lA . This shows that lA is a λ -system for any A P l ♣A q.
Now suppose A P A . Then A ❳ B P A ❸ l ♣A q for every B in the π -system A . Hence,
B P lA and thus lA ❹ A . But since l ♣A q is the smallest λ -system containing A , then
we have lA ❹ l ♣A q. Therefore, A ❳ B P l ♣A q for every B P l ♣A q. Finally, let B P l ♣A q
and consider lB ✏ tA ❸ Ω : A ❳ B P l ♣A q✉. Using our result before, we conclude that
lB is a λ -system and lB ❹ A . But this means A ❳ B P l ♣A q for every A P l ♣A q. Since
B P l ♣A 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
❸ B. Then, σ ♣A q ❸ B.
B a λ -system of subsets of Ω such that A
Proof. Let Ω be a nonempty set and A , B be π -system and λ -system of Ω, respec-
❸ B. Then, by definition of l ♣A q as the smallest λ -system containing A , we have l ♣A q ❸ B. From the previous lemma, we know that l ♣A q is a
σ -algebra, and since A ❸ l ♣A q, we have σ ♣A q ❸ l ♣A q. This completes the proof,
since l ♣A q ❸ B.
tively, such that A
Definition 2.1.4 (Measure). Let ♣Ω, F q be a measurable space. If µ : F
R is the extended nonnegative real number line R
ÑR
, where
✏ r0, ✽s, satisfies the following
axioms
(i) (Null empty set) µ ♣❍q ✏ 0
(ii) (Countable additivity) If A1 , ..., An , ... is a sequence of pairwise disjoint sets in
➈
F , then µ ♣ ✽
n✏1 An q ✏
➦ ✽ µ ♣A q,
n
n✏1
8
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
then we call µ a measure on ♣Ω, F q and ♣Ω, F , µ q a measure space. If µ ♣Ωq ✏ 1,
then we call µ a probability measure and ♣Ω, F , µ q a probability space.
Definition 2.1.5 (Outer Measure). Let Ω be an arbitrary set. A set function µ ✝ : 2Ω Ñ
R is called an outer measure on Ω if it satisfies the following axioms
(i) (Null empty set) µ ✝ ♣❍q ✏ 0.
(ii) (Monotonicity) If A1 , A2 P 2Ω such that A1 ❸ A2 , then µ ✝ ♣A1 q ↕ µ ✝ ♣A2 q.
➈
(iii) (Countable subadditivity) If A1 , ..., An , ... P 2Ω , then µ ✝ ♣ ✽
n✏1 An q ↕
➦✽
n✏1 µ
✝ ♣An q.
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., µ ✝ ♣A1 ❨ A2 q ✏ µ ✝ ♣A1 q
➈
µ ✝ ♣A2 q for every A1 , A2 P
q ✏ ➦Nn✏1 µ ✝♣Anq
➈ A q➙
for every N P N and pairwise disjoint sets A1 , ..., An , ... P 2Ω so that µ ✝ ♣ ✽
n✏1 n
➦✽ µ ✝♣A q. Using this result together with the countable subadditivity of µ ✝, we
n
n✏1
✝
2Ω such that A1 ❳ A2 ✏ ❍, then we have µ ✝ ♣ ✽
n✏1 An q ➙ µ ♣
➈N
n✏1 An
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
µ ✝ ♣ T q ✏ µ ✝ ♣ T ❳ Aq
µ ✝ ♣T ❳ Ac q for any set T
P 2Ω .
Definition 2.1.6 (Covering class of a set). Let Ω be an arbitrary set and A
❸ 2Ω be a
collection of subsets of Ω such that
(i) (Empty set) ❍ P A .
(ii) (Countable covering) Ω has a countable covering in A , i.e., there exist a sequence A1 , ..., An , ... P A such that
➈✽
n✏1 An
then we call A as the covering class of Ω
9
✏ Ω,
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 , ... P A such that
➈✽
n✏1 An
❹ 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
µ ✝ : 2Ω Ñ r0,
Ñ r0, ✽s such that δ ♣❍q ✏ 0. Then, the set function
✽s defined by
✰
★➳
✽
✽
↕
µ ✝ ♣Aq ✏ inf
δ ♣An q : A1 , ..., An , ... P A and
An ❹ A
n✏1
n✏1
is an outer measure on Ω based on δ .
Proof. It is obvious that µ ✝ ♣Aq P r0,
and hence µ ✝ ♣❍q ✏ 0.
✽s for every A P 2Ω and µ ✝♣❍q ↕ δ ♣❍q ✏ 0
Next, if A1 , A2 P 2Ω such that A1 ❸ 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 µ ✝ ♣A1 q and
µ ✝ ♣A2 q, respectively. Then, M ❸ N and consequently µ ✝ ♣A1 q ✏ inf M ↕ inf N ✏ µ ✝ ♣A2 q
which proves the monotonicity of µ ✝ . Last, let A1 , ..., An
→ 0, then
➈ A ❹ A and
for each n P N there exist a sequence ♣An,k qkPN in A such that ✽
n
k✏1 n,k
➦✽ δ ♣A q ↕ µ ✝♣A q ε . Therefore, ➈✽ ♣➈✽ A q ❹ ➈✽ A which implies
n
n,k
k✏1
n✏1
k✏1 n,k
n✏1 n
2
P 2Ω.
Given ε
n
that
✽
✽
✽
✽ ➳
✽
➳
↕
➳
➳
ε
✝
✝
µ ♣ An q ↕
r δ ♣An,k qs ↕ ♣µ ♣Anq 2n q ✏ µ ✝♣Anq ε
n✏1
n✏1
n✏1 k✏1
n✏1
and since ε
→ 0 is arbitrary, the conclusion follows.
Definition 2.1.7 (Lebesgue measure). Let I0 be the collection of ❍ and all open intervals in R. Let δ be a set function on I0 , i.e., δ : I0 Ñ r0,
✽s such that δ ♣❍q ✏ 0 and
➈ I q ✏ ➦ ✽ δ ♣I q.
for any sequence ♣In qnPN of disjoint intervals in R, we have δ ♣ ✽
n
n✏1 n
n✏1
If I is an interval in R with endpoints a, b P R where a ➔ b, then we define δ ♣I q ✏ b ✁ a
and if I is an infinite interval in R, we define δ ♣I q ✏ ✽. Define the Lebesgue outer
10
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
measure on R, µ ✝ : 2R Ñ r0,
µ ✝ ♣Aq ✏ inf
★✽
➳
n✏1
✽s, as follows
δ ♣An q : A1 , ..., An , ... P I0 and
✽
↕
n✏1
✰
An ❹ A
We call the restriction of µ ✝ to the collection of all Lebesgue measurable (i.e. measurable 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 µ ✝ ♣Aq ✏ 0.
Proof. First we will show that every singleton has measure zero. Let x P R and ε
→
0. Then ♣x ✁ ε2 , x ε2 q is an open cover of tx✉. Hence, we have µ ✝ ♣tx✉q ↕ δ ♣♣x ✁
ε
ε
✝
2,x
2 qq ✏ ε . Since ε → 0 is arbitrary, it follows that µ ♣tx✉q ✏ 0. Now let A be a
➈
countable subset of R, then A ✏ nPN txn ✉ is a countable union of null sets An ✏ txn ✉
➦
and therefore µ ✝ ♣Aq ↕ nPN µ ✝ ♣An q ✏ 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 ⑨
♣a ✁ ε2 , b ε2 q P I0, i.e., ♣a ✁ ε2 , b ε2 q is an open cover in I0 of I. Hence, we have
µ ✝ ♣I q ↕ δ ♣♣a ✁ ε2 , b ε2 qq ✏ δ ♣I q ε . Since ε is arbitrary, then µ ✝ ♣I q ↕ δ ♣I q. On the
➦
other side, let ♣In qnPN ⑨ I0 be a covering sequence of I. We will show that nPN δ ♣In q ➙
➦
δ ♣I q. If there exist k such that Ik P ♣In qnPN is an infinite interval, then nPN δ ♣In q ➙
δ ♣Ik q ✏ ✽ → δ ♣I q and we are done. So we assume that each In is a finite open
interval. Let ♣Jn qnPN be a subsequence of ♣In qnPN such that Jn ❳ I ✘ ❍ for every n and
➦
➦
Jk ❶ Jl if k ✘ l. Then nPN δ ♣Jn q ↕ nPN δ ♣In q. Since I is compact and ♣Jn qnPN is
an open cover of I, then there exist a finite subcover ♣Jn qN
k✏1 ⑨ ♣Jn qnPN of I. Assume
that Jn ✏ ♣an , bn q with an ↕ ... ↕ an . Obviously, if an ✏ an for i ✘ j, then we have
Jn ⑨ Jn or vice versa which contradicts the fact that Jk ❶ Jl if k ✘ l. Therefore, we
have an ➔ ... ➔ an . Next we will show that am 1 ➔ bm for every m P tn1 , ..., nN ✁1 ✉.
k
i
i
i
i
1
N
i
j
1
N
11
j
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
➙ bm for some m. Since Jn ❳ I ✘ ❍ for every n, then there exist
xm P Jm ❳ I and xm 1 P Jm 1 ❳ I such that am ➔ xm ➔ bm ↕ am 1 ➔ xm 1 ➔ bm 1 .
Since I is an interval and xm , xm 1 P I, we have rxm , xm 1 s ⑨ I. Consider the following
➈ J . If b ➔ a , then
two possibilities. If bm ✏ am 1 , then bm P I but bm ❘ N
m
m 1
i✏1 n
rbm, am 1s ⑨ I but rbm, am 1s ❶ ➈Ni✏1 Jn which contradicts the fact that ♣Jn qNi✏1 is an
open cover of I. Thus we have am 1 ➔ bm for every m P tn1 , ..., nN ✉. So we have the
If not, then am
1
i
i
i
following inequalities
➳
nPN
δ ♣In q ➙
➳
nPN
δ ♣Jn q ➙
➳N
i✏1
δ ♣Jni q ✏ ♣bn1 ✁ an1 q
➙ ♣ an ✁ an q
➙ bn ✁ an
➙ b✁a
✏ δ ♣I q,
2
1
N
i.e., δ ♣I q is a lower bound of
➦
nPN δ
...
♣bn ✁ ✁ an ✁ q ♣bn ✁ an q
...
♣an ✁ an ✁ q ♣bn ✁ an q
N 1
N
N
N 1
N 1
N
N
N
1
♣Inq so that µ ✝♣I q ➙ δ ♣I q and the conclusion
follows. Second, let I ✏ ♣a, 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
µ ✝ ♣♣a, bqq ↕ µ ✝ ♣ra, bsq ↕ µ ✝ ♣ta✉q
µ ✝ ♣♣a, bqq
µ ✝ ♣tb✉q ✏ µ ✝ ♣♣a, bqq
which implies that µ ✝ ♣♣a, bqq ✏ µ ✝ ♣ra, bsq ✏ δ ♣ra, bsq ✏ δ ♣♣a, bqq. Third, if I is a finite
✏ ♣a, bs, then we have µ ✝♣♣a, bqq ↕ µ ✝♣♣a, bsq ↕ µ ✝♣♣a, bqq
µ ✝ ♣tb✉q ✏ µ ✝ ♣♣a, bqq so that µ ✝ ♣♣a, bsq ✏ µ ✝ ♣♣a, bqq ✏ δ ♣♣a, bqq ✏ δ ♣♣a, bsq. Clearly
for I ✏ ra, bq we also have µ ✝ ♣ra, bqq ✏ δ ♣ra, bqq. Last, if I is an infinite interval, say
of the form I ✏ ♣a, ✽q, then we have µ ✝ ♣♣a, ✽qq ➙ µ ✝ ♣♣a, nqq ✏ δ ♣♣a, nqq ✏ n ✁ a
for any n → a so that µ ✝ ♣♣a, ✽qq ✏ ✽ ✏ δ ♣♣a, ✽qq. Similar arguments also hold
interval of the form I
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PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
for any other infinite interval I.
Definition 2.1.8 (Translation Invariance). Let Ω be a vector space, ♣Ω, F , µ q a measure space, and A
x ✏ ta
x : a P A✉ for any A P F 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 P Ω, A x has the same measure
as A itself.
(iii) ♣Ω, F , µ q 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 ♣In qnPN of A in I0 , then we have
➈
xq ✏ ♣ ✽
n✏1 In q
x ❹ A and hence
➈ ✽ ♣I
n✏1 n
➦ ✽ δ ♣I q ✏ ➦ ✽ δ ♣I x q ➙ µ ✝ ♣A x q.
n
n
n✏1
n✏1
Since
♣InqnPN is an arbitrary open interval cover of A, then µ ✝♣Aq ➙ µ ✝♣A xq from the
➦ δ ♣I q. Conversely, by applying the similiar
definition of µ ✝ as the infimum of ✽
n
n✏1
method to the set A x and its translate ♣A xq✁ x ✏ A, we have the reverse inequality
µ ✝ ♣A xq ➙ µ ✝ ♣Aq 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
µ ✝ ♣T ❳♣A
xqq
µ ✝ ♣T ❳♣A
xqc q ✏ µ ✝ ♣tT ❳♣A
13
xq✉✁ xq
µ ✝ ♣tT ❳♣A
xqc ✉✁ xq
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
✏ µ ✝♣♣T ✁ xq❳ Aq
✏ µ ✝ ♣T ✁ x q
✏ µ ✝ ♣T q
µ ✝ ♣♣T ✁ xq❳ Ac 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
µ ♣A
x q ✏ µ ✝ ♣A
xq ✏ µ ✝ ♣Aq ✏ µ ♣Aq,
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 ♣Ω, F , µ q 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 ♣Ω, F0 , µ0 q such that F0
❹F
and µ0
✏µ
on F , then we say that ♣Ω, F0 , µ0 q is a complete extension of ♣Ω, F , µ q. If it is the
smallest complete extension of ♣Ω, F , µ q, i.e, if for any complete extension ♣Ω, F1 , µ1 q
of ♣Ω, F , µ q we have F1 ❹ 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
P 2R, we have 0 ↕ µ ✝♣T ❳ Aq ↕ µ ✝♣Aq ✏ 0
and hence µ ✝ ♣T ❳ Aq ✏ 0. Therefore, µ ✝ ♣T q ↕ µ ✝ ♣T ❳ Ac q ✏ µ ✝ ♣T ❳ Aq µ ✝ ♣T ❳ Ac q
measurable. Clearly, for any testing set T
so that A is Lebesgue measurable. Next, assume that A has Lebesgue measure zero and
B ⑨ A, then µ ✝ ♣Bq ↕ µ ✝ ♣Aq ✏ µ ♣Aq ✏ 0 so that B is Lebesgue measurable.
Since every interval is Lebesgue measurable and every open set is a union of count14
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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 ♣X, A q and ♣Y, B q be measurable spaces.
A function f : X
Ñ Y is called measurable (B ✁ A -measurable) if for all B P B, we
have f ✁1 ♣Bq P A .
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 ♣X, A q, ♣Y, B q be measurable spaces and G
❸ 2Y be the generator
of B, i.e., B ✏ σ ♣G q. Then a function f : X Ñ Y is measurable if and only if f ✁1 ♣Gq P
A for every G P G .
Proof. Clearly if f is measurable, then f ✁1 ♣Gq P A for every G P G . Define M
✥
✭
✏
M P B : f ✁1 ♣M q P A . It is easy to show that M is a σ -algebra as follows: since
f ✁1 ♣Y q ✏ X
P A , then Y P M . If M P M , then f ✁1♣Mq P A , and hence ♣ f ✁1♣Mqqc ✏
f ✁1 ♣M c q P A so that M c P M . Finally, if M1 , ..., Mn , ... P M , then f ✁1 ♣Mi q P A for
➈
➈
every i P N which implies that ni✏1 f ✁1 ♣Mi q ✏ f ✁1 ♣ ni✏1 ♣Mi qq P A . Therefore, since
G ❸ M , then σ ♣G q ❸ σ ♣M q, i.e, B ❸ 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
ak
P R and Ek ✏ φ ✁1♣tak ✉q.
✏ ➦nk✏1 ak 1E
k
where
This definition of simple functions play a fundamental
role in measure theory. This is also an implication of the following theorem.
15
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Theorem 2.1.7 (Approximation by simple functions). If f : X
ÑR
is a nonnegative
measurable function, then there is a monotone increasing sequence of nonnegative
simple functions ϕn : X
ÑR
such that ϕn Ñ f pointwise as n Ñ
✽.
Proof. For each n P N, we divide the interval r0, 2n q in the range of f into 22n subinter-
vals of width 2✁n as follows. Define Ik,n ✏ rk2✁n , ♣k
1q2✁n q for k ✏ 0, 1, ..., 22n ✁ 1
✽s. We also define Ek,n ✏ f ✁1♣Ik,nq and En ✏ f ✁1♣Inq. Then, it is easy to
➦
see that the sequence of simple functions ♣ϕn qnPN where ϕn ✏ k2✏✁01 k2✁n 1E
2n 1 E
and In ✏ r2n ,
n
k,n
n
satisfies the required properties.
Note that every function f can be written as a sum of two positive functions,
✏ f ✁ f ✁ where f ✏ max t f , 0✉ and called the positive part of f and f ✁ ✏
max t✁ f , 0✉ and called the negative part of f . This means that we can prove that some
i.e., f
properties hold for f just simply by proving that it is true for the corresponding simple
functions.
Definition 2.1.12 (Lebesgue Integral). Let ♣Ω, F , µ q be a measure space. The Lebesgue
integral over Ω of a measurable simple function φ : Ω Ñ R is defined as
➺
φ dµ
Ω
✏
➺ ➳
n
Ω k✏1
ak 1Ek d µ
✏
n
➳
k✏1
ak µ ♣Ek q.
If f : Ω Ñ R is a nonnegative measurable function, then the Lebesgue integral of f
➩
✥➩
✭
✏ sup Ω φ d µ : φ is simple and 0 ↕ φ ↕ f . Finally, for
Ω
any measurable function f : Ω Ñ R ✏ R ❨t✁✽, ✽✉, the Lebesgue integral of f over
➩
➩
➩
Ω is defined as Ω f d µ ✏ Ω f d µ ✁ Ω f ✁ d µ .
over Ω is defined as
f dµ
Theorem 2.1.8 (Monotone Convergence Theorem). Let ♣ fn qnPN be a sequence of non-
negative measurable functions in a measure space ♣Ω, F , µ q which increasing pointwise to f , then
➺
f dµ
Ω
✏
➺
lim fn d µ
Ω nÑ
✽
16
✏ nÑlim✽
➺
fn d µ .
Ω
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Proof. Since fn is measurable and increasing pointwise to f , then f is measurable.
➩
➩
↕ f , then Ω fnd µ ↕ Ω f d µ . Take any c P ♣0, 1q and fixed a simple
function 0 ↕ ϕ ↕ f . Let An ✏ tx P Ω : fn ♣xq ➙ cϕ ♣xq✉. It is clear that An is increasing.
➈
To show that X ✏ nPN An , let x P Ω. Then
Also, since fn
(i) If ϕ ♣xq ✏ 0, then x P An for any n P N.
(ii) If ϕ ♣xq → 0, from the fact that f ♣xq ➙ ϕ ♣xq → cϕ ♣xq, then there exist n P N such
that fn ♣xq ➙ t ϕ ♣xq.
This means that x P An . For any A P F , set v♣Aq ✏
where ϕ
✏ ➦mn✏1 yn1E
n
➩
✏ c ➦mn✏1 yn µ ♣A ❳ Enq
A cϕ d µ
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
➺
cϕ d µ
Ω
✏ v♣ Ωq ✏ v♣
↕
nPN
An q ✏ lim v♣An q ✏ lim
nÑ
✽
nÑ
✽
↕ nÑlim✽
↕ nÑlim✽
That is, c
limnÑ ✽
➩
Ω ϕdµ
➩
↕ limnÑ
Ω fn d µ .
✽
➩
Ω fn d µ .
➺
cϕ d µ
➺
An
➺
An
fn d µ
fn d µ .
Ω
Since c P ♣0, 1q is arbitrary, we have
➩
Ω ϕdµ
↕
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 ♣Ω, F , µ q
be a measure space and f : A③N Ñ R be a measurable function on A③N where A, N P
F , A ❹ N, and N is a null set. We write
➩
A
17
f d µ for
➩
A
frd µ where fr is a nonnegative
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
extended real valued measurable function defined on A by setting
✩
✬
✫
rf ♣xq ✏ f ♣xq
✬
✪0
if x P A③N
if x P N
.
➩ frd µ ✏ ➩ frd µ ➩ frd µ ✏ ➩ frd µ .
A
A③N
N
A③N
➩
The definition above of A f d µ for a function f which is defined only a.e. (almost
➩ f dµ.
everywhere) on A is for convenience of not having to write
It is a fact that
A③N
Theorem 2.1.9 (Fatou’s lemma). Let ♣Ω, F , µ q be a measure space and ♣ fn qnPN be an
arbitrary sequence of nonnegative extended real valued measurable functions on a set
A P F , then we have
➺
lim inf fn d µ
A nÑ
In particular, if f
✏ limnÑ
✽
↕ lim
inf
nÑ ✽
➺
fn d µ .
A
✽ fn exist a.e. on A, then
➺
f dµ
A
Proof. By definition, lim infnÑ ✽ fn
↕ lim
inf
nÑ ✽
➺
✏ limnÑ
fn d µ .
A
✽ infk➙n fk . Clearly, ♣infk➙n fk qnPN is
increasing and hence by the monotone convergence theorem
➺
➺
➺
lim inf fk d µ ✏ lim
inf fk d µ
lim inf fn d µ ✏
nÑ ✽ A k➙n
A nÑ ✽ k➙n
A nÑ ✽
➺
✏ lim
inf inf fk d µ
nÑ ✽ A k➙n
➺
↕ lim
inf
nÑ ✽
Last, if f
✏ limnÑ
✽ fn exist a.e. on A, then f
✏ lim infnÑ
fn d µ
A
✽ fn a.e. on A. Thus the
conclusion follows by Definition 2.12.
Theorem 2.1.10 (Lebesgue dominated convergence theorem). Let ♣Ω, F , µ q be a mea18
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Ñ R be a sequence of measurable functions such that fn Ñ f
pointwise as n Ñ ✽. Suppose that there exist an integrable function g such that ⑤ fn ⑤ ↕
➩
g for every n P N. Then fn and f are also integrable and limnÑ ✽ Ω ⑤ fn ✁ f ⑤d µ ✏ 0.
sure space and fn : X
Proof. Obviously fn and f are integrable. Also, 2g ✁⑤ fn ✁ f ⑤ is nonnegative and measurable. By Fatou’s lemma
➺
lim inf♣2g ✁⑤ fn ✁ f ⑤qd µ
Ω nÑ
✽
➺
↕ lim
inf ♣2g ✁⑤ fn ✁ f ⑤qd µ
nÑ ✽
Ω
Since fn Ñ f , the left hand side of the inequality above is just
➩
Ω 2gd µ
and the right
hand side is equal to
✂➺
lim inf
nÑ
✽
Ω
2gd µ ✁
✡
➺
Ω
⑤ f n ✁ f ⑤d µ ✏
✏
➺
Ω
Ω
2gd µ ✁ lim inf
nÑ
✂➺
✽
2gd µ ✁ lim sup
nÑ
Ω
✂➺
✽
⑤ f n ✁ f ⑤d µ
Ω
✡
⑤ f n ✁ f ⑤d µ
✡
➔ ✽, we may cancel the above inequality to obtain
✟
➩
➩
Ω ⑤ f n ✁ f ⑤d µ ↕ 0 and so limnÑ ✽ Ω ⑤ f n ✁ f ⑤d µ ✏ 0.
Using the fact that
lim supnÑ ✽
➩
➺
Ω 2gd µ
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 ⑤ fn ⑤ ↕ g a.e.
(ii) By the triangle inequality, it is clear that limnÑ ✽
➩
Ω
fn d µ
✏
➩
Ω
f dµ.
Theorem 2.1.11 (Beppo-Levi). Let ♣Ω, F , µ q be a measure space and fn : Ω Ñ r0,
be a sequence of nonnegative measurable functions. Then
➺ ➳
Ω nPN
fn d µ
✏
➳➺
nPN Ω
19
fn d µ .
✽s
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Proof. Let gN
✏ ➦Nn✏1 fn and g ✏ ➦nPN fn.
By the monotone convergence theorem,
we have
➺
gd µ
Ω
✏
➺
lim gN d µ
Ω NÑ
✽
✏ NÑlim✽
➺
✏ NÑlim✽
gN d µ
Ω
➳N ➺
n✏1 Ω
fn d µ
✏
✽➺
➳
n✏1 Ω
fn d µ .
B. Tools from Functional Analysis
Definition 2.2.1 (Normed space). A normed space is a vector space X over a field
P tR, C✉ equipped with a function ⑥ ☎ ⑥ : X Ñ R, called a norm, such that for every
x, y P X and λ P F the following axioms hold
F
(i) (Nullity) ⑥x⑥ ✏ 0 implies x ✏ 0.
(ii) (Homogeinity) ⑥λ x⑥ ✏ ⑤λ ⑤⑥x⑥.
(iii) (Triangle inequality) ⑥x
y⑥ ↕ ⑥ x ⑥
⑥ y⑥ .
From these axioms, we can easily deduce that ⑥x⑥ ➙ 0 and ⑥x⑥ ✏ 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) d ♣x, yq ✏ 0 if and only if x ✏ y.
(ii) (Symmetry) d ♣x, yq ✏ d ♣y, xq.
(iii) (Triangle inequality) d ♣x, yq ↕ d ♣x, zq
d ♣z, yq.
From the third axiom, we also can easily deduce that d ♣x, yq ➙ 0 for every x, y P X. Also
notice that every normed space is metrizable, i.e., we can generate a metric d ♣x, yq :✏
⑥x ✁ y⑥ for every x, y P X in a normed space.
20
This is the standard metric that will be
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
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
♣xnqnPN 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 d ♣xn , xq ➔ ε for every n ➙ N. A sequence ♣xn qnPN in X is called a
Cauchy sequence if for every ε → 0, there exist N P N such that d ♣xn , xm q ➔ 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 ✏ n1 , n P N in the metric space of real interval ♣0, 1q with the
usual metric d ♣x, yq ✏ ⑤x ✁ y⑤ which is Cauchy but not convergent to an element in
♣0, 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, z P X and λ
①☎, ☎② : X ✂ X Ñ F that satisfies the following axioms for any
P F.
(i) (Conjugate Symmetry) ①x, y② ✏ ①y, x②.
(ii) (Distributive) ①x
y, z② ✏ ①x, z②
①y, z②.
(iii) (Homogeneity) ①λ x, y② ✏ λ ①x, y②.
(iv) (Positive definiteness) ①x, x② ➙ 0 and ①x, x② ✏ 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., ⑥x⑥ ✏ ①x, x② 2 for some inner product ①☎, ☎②
1
defined on the normed space H , we call H a Hilbert space.
21
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Definition 2.2.6 (Orthonormal Basis of Hilbert Spaces). Let ♣ei qiPI be a collection of
vectors in H . We call ♣ei qiPI an orthonormal basis of H if ①ei , e j ② ✏ 0 for i ✘ j and
①h, ei② ✏ 0 for every i P I implies that h ✏ 0.
Denote by Cb ♣H q the space of all continuous bounded mappings from H into R
endowed by the norm ⑥ f ⑥0 ✏ supxPH ⑤ f ♣xq⑤ for any f
P Cb♣H q.
Proposition 2.2.1. Cb ♣H q is a Banach space.
Proof. Let ♣ fn qnPN be a Cauchy sequence in Cb ♣H q. Since ⑤ fn ♣xq ✁ fm ♣xq⑤ ↕ ⑥ fn ✁
fm ⑥0 , the sequence ♣ fn ♣xqqnPN 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 ♣xq ✏ limnÑ ✽ fn ♣xq is an R-
valued function. Now we will show that the limit f is a bounded function as fol-
→ 0 and N be a positive integer such that ⑥ fn ✁ fm⑥0 ➔ ε for all n, m ➙ N.
Then, the inequality ⑤ f ♣xq⑤ ↕ ⑤ f ♣xq ✁ fN ♣xq⑤ ⑤ fN ♣xq⑤ ↕ ⑥ f ✁ fN ⑥0 ⑥ fN ⑥0 holds for
all x P H . But since for n ➙ N, ⑤ fn ♣xq✁ fN ♣xq⑤ ➔ ε for every x, then ⑤ f ♣xq✁ fN ♣xq⑤ ✏
limnÑ ✽ ⑤ fn ♣xq ✁ fN ♣xq⑤ ↕ ε so that ⑥ f ✁ fN ⑥0 ✏ supxPH ⑤ f ♣xq ✁ fN ♣xq⑤ ↕ ε . This
lows: Let ε
shows that the function f is bounded. Let x0 in H . By the continuity of fN , choose
→ 0 such that ⑤ fN ♣xq ✁ fN ♣x0q⑤ ➔ ε whenever ⑥x ✁ x0⑥ ➔ δ . Then if ⑥x ✁ x0⑥ ➔ δ ,
we have ⑤ f ♣xq ✁ f ♣x0 q⑤ ↕ ⑤ f ♣xq ✁ fN ♣xq⑤ ⑤ fN ♣xq ✁ fN ♣x0 q⑤ ⑤ fN ♣x0 q ✁ f ♣x0 q⑤ ↕ ⑥ f ✁
fN ⑥0 ⑤ fN ♣xq ✁ fN ♣x0 q⑤ ⑥ fN ✁ f ⑥0 ➔ ε ε ε ✏ 3ε . This shows the continuity of
f . To finish the proof we need to show fn converges in norm, i.e. ⑥ f ✁ fn ⑥0 Ñ 0
as n Ñ ✽. This is clear by using the triangle inequality as follows: ⑥ f ✁ fn ⑥0 ↕
⑥ f ✁ fN ⑥0 ⑥ fN ✁ fn⑥0 ↕ ε ε ✏ 2ε so that ⑥ f ✁ fn⑥0 Ñ 0.
δ
Theorem 2.2.1 (The orthogonal decomposition theorem). Let H be a Hilbert space
and S ❸ H a closed subspace of H . Then the orthogonal complement S❑ defined by
S❑ ✏ tx P X : ①x, y② ✏ 0 for every y
P S✉ is also a closed subspace of H
be represented as the direct sum of S and S❑ , i.e., H ✏ S ❵ S❑ .
22
and H can
PLAGIAT MERUPAKAN TINDAKAN TIDAK TERPUJI
Proof. It is clear that S❑ is a vector sub