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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.


Course coordinator: Maria Roginskaya

Course literature

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed. Wiley 1999, Chapters 1-3. (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


Preliminary program (some adjustments can be done during the course)





F pp.19-22
JJ 3.1,2,5

Why can we not measure all sets? Concept of s-algebra. p- and d-systems. Dynkin lemma


JJ 3.7 + Thm 3.10, F 1.3,4

Outer measure, Carathéodory’s theorem


F 2.1,2 up to (but not including) Thm 2.14

JJ 5 till Thm 5.6

Measurable functions, integration of non-negative functions.


F Thm 2.14-20, F 2.3 till p.55

Monotone convergence theorem, integration of complex valued functions, Lebesgue’s dominated convergence theorem


F 2.4

Modes of convergence


F p.22+ F2.5

Product s-algebras, product measures, Fubini-Tonelli theorem


JJ p.16,pp.21-22,24-25

Lebesgue-Stieltjes measures, Expectations, Independent random variables


JJ pp.9-10, 26-28

Borell-Cantelli I and II, Kolmogorov’s 0-1 law


F 3.1

Signed measures, Jordan-Hahn decomposition theorem


F 3.2

Radon-Nikodym theorem


Lebesgue’s decomposition theorem


F pp.95-96

3-times covering lemma, Hardy-Littlewood maximal function, maximal theorem


F pp.97-100

Lebesgue’s differentiation theorem


F pp.101-104

Functions of bounded variation


F pp. 105-107

Fundamental theorem of calculus and integration by parts for Lebesgue integral


Presentation of self-study projects



Riesz products and other singular measures




Recommended excercises:

F 1.2.4

1.4.17, 1.4.18, 1.4.24

F 2.2.13, 2.2.15

F 2.4.33-39,42

F 2.5.45,(46),48

Read examples on JJ pp.21-22

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


During the course a self-study 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 sigma-algebras for a finite set

·         Examples of sigma-algebras on infinite sets (for example finite/co-finite sets, etc.)

·         Different types of convergency and connection between them

·         Approximation by continuous functions

·         More covering lemmas/theorems


Oral examination + the presentation of the self-study 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.