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Welcome to the course.
Here
is a bit more careful calculation for the proof of a.e.
convergence of a subsequence of a sequence convergent in measure (from 9/9)
Here
is the alternative prove of Hahn Decomposition theorem.
Here
are abbreviated notes for the lecture on Riesz
products
Here
are questions to the oral exam.
Examination dates are
(to those who signed for them): 26/10 from 10.00 and 27/10 from 10.00 in MVL15.
Those who take the examination on 27/10 should get their question numbers from Lotta Fernström (MVL3030) and
then go to MVL15 to prepare.
If you haven’t signed on
the list for the examination you should contact the examiner for another time
for the examination.
The schedule for the course can be found via the link to webTimeEdit
top of the page.
Teachers
Course coordinator: Maria Roginskaya
Course literature
Gerald B. Folland, Real Analysis: Modern Techniques and Their
Applications, 2nd ed. Wiley 1999, Chapters 13. (abbreviated F below) (Warning:
At many bookstores, they sell what may appear to be a paperback version of the
book, but unfortunately it is only something of a glossary. The real book is
available only in hardback.) You also
will need this book for the next course MMA120 Functional Analysis
Johan Jonasson, (abbreviated JJ below) notes can be downloaded here
Programme
Preliminary program (some adjustments can be done during the course)
Date 
Chapter 
Contents 
9/1 
F pp.1922 
Why can we not measure all sets?
Concept of salgebra. p and
dsystems. Dynkin lemma 
9/2 
JJ 3.7 + Thm 3.10, F 1.3,4 
Outer measure, Carathéodory’s
theorem 
9/5 
F 2.1,2 up to (but not including) Thm 2.14 JJ 5 till Thm 5.6 
Measurable functions, integration
of nonnegative functions. 
9/8 
F Thm 2.1420, F 2.3 till p.55 
Monotone convergence theorem,
integration of complex valued functions, Lebesgue’s dominated convergence
theorem 
9/9 
F 2.4 
Modes of convergence 
9/12 
F p.22+ F2.5 
Product salgebras,
product measures, FubiniTonelli theorem 
9/15 
JJ p.16,pp.2122,2425 
LebesgueStieltjes
measures, Expectations, Independent random variables 
9/16 
JJ pp.910, 2628 
BorellCantelli I and II, Kolmogorov’s 01 law 
9/19 
F 3.1 
Signed measures, JordanHahn
decomposition theorem 
9/22 
F 3.2 
RadonNikodym
theorem 
9/23 
Lebesgue’s decomposition theorem 

9/26 
F pp.9596 
3times covering lemma,
HardyLittlewood maximal function, maximal theorem 
9/29 
F pp.97100 
Lebesgue’s differentiation theorem 
9/30 
F pp.101104 
Functions of bounded variation 
10/6 
F pp. 105107 
Fundamental theorem of calculus
and integration by parts for Lebesgue integral 
10/7,10,13,14 
Presentation of selfstudy
projects 

10/17 


10/20 
Summary 
Recommended
excercises:
F 1.2.4
1.4.17, 1.4.18, 1.4.24
F 2.2.13, 2.2.15
F 2.4.3339,42
F 2.5.45,(46),48
Read examples on JJ
pp.2122
F 3.1.1, 2, 3, 6
F 3.2.8,10, 11, 12*, 13
F 3.2.16, 17
F 3.4.22, 23
F 3.4.24, 25, 26
F 3.5.30, 31, 32
F 3.5.33, 40, 41
Course requirements
90 credit points in
Mathematics, recommended knowledge include Real and Complex Analysis
Assignments
During the course a
selfstudy project should be done and presented.
Examples for the topic:
·
Construction of Borel
sets (via ordinals)
·
Original construction of Lebesgue measure
(via inner and outer capacities)
·
Finding all sigmaalgebras for a finite
set
·
Examples of sigmaalgebras on infinite sets
(for example finite/cofinite sets, etc.)
·
Different types of convergency
and connection between them
·
Approximation by continuous functions
·
More covering
lemmas/theorems
Examination
Oral examination + the
presentation of the selfstudy project
The date of examination
is according to agreement with the examiner
Examination procedures
In Chalmers Student Portal you can read about when
exams are given and what rules apply on exams at Chalmers.
At the exam, you should be able to show valid identification.
Before the exam, it is important that you report that you want to take the
examination to the course coordinator.