## Latest news

Welcome to the course! The schedule for the course can be found in TimeEdit.

The Reading Projects are with the following topics:

- Construction of Borel sets
- Classification of σ-algebras on a finite set
- Hausdorff measures
- Convergence of different types and on different measures

The oral examination will be held on 16/10, 25/10, and 26/10, please write to the examiner at which date you intend to have the examination (and if you would like to start before 9.20). (Notice, that 26/10 is closed to be overbooked, if you can choose another date, please, do so.)

The theoretical part of the examination will consist of two questions out of the following list.

## Teachers

Course coordinator:**Maria Roginskaya**

Guest Lecturer: **Johan Jonasson**

## 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

#### Lectures

Day |
Sections | Contents |
---|---|---|

28/8 | F pp.19-22 JJ 3.1,2,5 |
Why can we not measure all sets? Concept of σ-algebra. π- and d-systems. Dynkin lemma |

31/8 | JJ 3.7 + Thm 3.10 F 1.3,4 |
Outer measure, Caratheodory's theorem |

1/9 | 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. |

4/9 | F Thm 2.14-20, F 2.3 till p.55 | Monotone convergence theorem, integration of complex valued functions, Lebesgue’s dominated convergence theorem |

7/9 | F 2.4 | Modes of convergence |

8/9 | F p.22-23+ F2.5 | Product σ-algebras, product measures, Fubini-Tonelli theorem |

11/9 | JJ p.16,pp.21-22,24-25 | Lebesgue-Stieltjes measures, Expectations, Independent random variables |

14/9 | JJ pp.9-10, 26-28 | Borell-Cantelli lemmas, I and II, Kolmogorov’s 0-1 law |

15/9 | F 3.1 | Signed measures, Jordan-Hahn decomposition theorem |

18/9 | F 3.2 | Radon-Nikodym theorem |

21/9 | Lebesgue’s decomposition theorem | |

22/9 | F pp.95-96 | 3-times covering lemma, Hardy-Littlewood maximal function, maximal theorem |

25/9 | F pp.97-100 | Lebesgue’s differentiation theorem |

28/9 | F pp. 101-104 | Functions of bounded variation |

29/9 | F pp. 105-107 | Fundamental theorem of calculus and integration by parts for Lebesgue integral |

6/10 | |
Presentations of reading projects |

9/10 | ||

12/10 | ||

13/10 |

#### Recommended exercises

F 1.2.4

F 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

The learning goals of the course can be found in the course plan.

## Examination

The course is examinated by a short reading project towards the end of the course, and an oral examination. Dates for the oral examination will be agreed on during the course.

## Examination procedures

In Chalmers Student Portal you can read about when exams are given and what rules apply on exams at Chalmers. In addition to that, there is a schedule when exams are given for courses at University of Gothenburg.

Before the exam, it is important that you sign up for the examination. If you study at Chalmers, you will do this by the Chalmers Student Portal, and if you study at University of Gothenburg, you sign up via GU's Student Portal, where you also can read about what rules apply to examination at University of Gothenburg.

At the exam, you should be able to show valid identification.

After the exam has been graded, you can see your results in Ladok by logging on to your Student portal.

**At the annual (regular) examination: **

When it is practical, a separate review is arranged. The date of the
review will be announced here on the course homepage. Anyone who can not
participate in the review may thereafter retrieve and review their exam
at the Mathematical
Sciences Student office. Check that you have the right grades and
score. Any complaints about the marking must be submitted in writing at
the office, where there is a form to fill out.

** At re-examination: **

Exams are reviewed and retrieved at the Mathematical
Sciences Student office. Check that you have the right grades and
score. Any complaints about the marking must be submitted in writing at
the office, where there is a form to fill out.