# TMA4225 Foundations of analysis: Fall 2011

The course will be taught by Harald Hanche-Olsen. Lectures will be given in English.

## Messages

- (2011-12-19) Today's exam problems, with my solutions are found under the “exam problems” link to the left. I have not had time to proofread the solution; I appreciate it if you report any errors or misprints to me.
- (2011-12-14) My office hours on Friday will be in the afternoon: From 13:00 to 16:00.
- (2011-12-12) I will hold office hours on Friday. Exact times yet to be decided. Sorry, I have no time to spare before then.
- (2011-12-12) Old exam problems are available. See menu in the left margin.
- (2011-12-07) The reading list is ready. See the bottom of this page.
- (2011-11-21) In this final week, I intend to convert the exercise hour into a lecture.

Note also the update to the mini-note on Littlewood's principles (below). - (2011-11-03) Results from the “midterm” test. On page 1 are the answers recorded, with the correct answers on top. Page 2 shows the result of awarding one point to each correct answer, summed and normalized to 20 points for a full answer (since the test is supposed to count 20%). Statistics on the bottom. Please note: I may decide to compute points differently. I have not decided that yet.
- (2011-10-28) Here is today's “midterm” test: unmarked and marked up with answers. Also, a somewhat sketchy justification of those answers.
- (2011-10-25) No exercises this week, because of the “midterm” test. Instead, I will be available in my office 12–13 on Thursday, 27 October
- (2011-10-25)
**What is covered by the “midterm” test?**

From Eugenia's notes (as slightly modified by me): Part I, sections 1–4.

Fremlin's volume 1: Chapters 11 (excluding 115) and 12.

Fremlin's volume 2: Nothing. - (2011-10-25) The “midterm” test will be during the lecture hours 08:15–10:00 on
**Friday, October 28**in rooms**F2**and**F3**(i.e., where the lectures are usually held plus a nearby room).**Remember to bring your student ID!**No aids are allowed except the so-called “exam calculator”, but I guarantee you that you will not find a use for it. The test will count 20% towards your final grade, but only in the positive direction. In particular, the test is not mandatory. If you don't take it, the final exam will just count 100%. Also, even an excellent result on the midterm test cannot save you from a failing grade, as the regulations insist on a passing grade on the final exam. Finally,**if you have a documented need for extra time or other special needs**, contact Inger Seehuus at the mathematics department office as soon as possible. - (2011-09-19) There will be no exercise session on Thursday, September 29 and no lecture on Friday, September 30 (I will be in Kragerø that day).
- (2011-08-31) The first exercises are (finally!) posted (see the menu on the left). I apologize for doing so very late; I will be quicker about it in the future.
- (2011-08-29) I have edited Eugenia's notes (part 1) a bit further. None of the changes are very serious, I think.

## Lectures and exercises

All activities in auditorium F3.

Lectures: Tuesdays 14–16 and Fridays 08–10.

Exercises: Thursdays 12–13.

First lecture on Tuesday, 23 August. First exercise session on Thursday, 1 September.

## Textbook

I plan to base my lectures on the first two volumes of D.H. Fremlin: Measure theory. Note that the books' TeX source code is available from the web site (linked below). I have compiled the code, and the result is here: volume 1, volume 2 [*]. It is also possible to order print copies directly from the publisher; see the author's web site, below. The Tapir book store has a few copies, too. (Now sold out, but more are coming.)

We will also use part 1 of some notes that Eugenia Malinnikova wrote when she had the course. I have made some small edits to the notes, in particular adapted the notation for sequences to match that of Fremlin's book. The result is here: Notes, part 1.

I have written up a short note on Littlewood's three principles (a5 (for the screen), a4 (for paper)). Updated 2011-11-21, with a proof of the Riemann–Lebesgue lemma

I may also use ideas from some other books (I'll list them later).

[*] According to the license for Fremlin's books, I must also distribute the source code, or offer to distribute it. However, I suggest you get it from the *author's site* if you can. If for some reason you cannot, email me, and I'll get it to you.

## Reading list

Here, at long last, is the official reading list.

Note | The notes by Eugenia Malinnikova, as amended by me (part 1) |
---|---|

Note | The note on Littlewood's three principles (a5, a4) |

Fremlin | “Starred” sections are excluded, unless explicitly mentioned The appendices 1A1–1A3 are useful, but overlap quite a bit with part of the above notes |

Ch 11 | 111–114 |

Ch 12 | 121–123 |

Ch 13 | 131A–D, 132A–C, 133A–G, 136A–B |

Ch 21 | 211A–D |

Ch 22 | 221–225 |

Ch 25 | 251A–K. 252A–H |

### Additional notes

Fremlin puts quite a bit of emphasis on functions that might not be defined everywhere, and which are measurable only when restricted to some co-negligible set (“virtually measurable”). This results in a more complicated theory, which we have largely neglected. In particular, we shall avoid such complication for the exam.

Likewise, I have only lectured a highly simplified version of sections **251–252** (product measures and the Fubini and Tonelli theorems), where we first assumed the measure spaces were finite – after which we extended the results to the σ-finite case. In this case, the two product measures defined in the book coincide (see **251K**), and this is the only case we considered in the course.