TMA4225 Foundations of analysis: Fall 2011


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


  • (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.


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.

2011-12-19, Harald Hanche-Olsen