TMA4225 Analysens grunnlag – Foundations of Analysis
Lecturers
- Office: room 1156, Sentralbygg 2
and
- Office: room 1129, Sentralbygg 2
Textbook
- Sheldon Axler: Measure, Integration & Real Analysis, Springer 2020. Click here for the pdf file (from NTNU accounts). A paperback version can be purchased from Springer for € 39.99. A free pdf is available from the author's homepage, or here. Axler has also written a supplement to his book, see Supplement.
Supplementary notes
- Notes on the Lebesgue integral – a quicker way to get the initial results for the Lebesgue integral (Axler section 3A) – updated 2025-09-22
- A uniqueness theorem and its consequences – latest version 2025-10-16: Translational and rotational invariance of Lebesgue measure, area and volume of spheres. The 2025-10-16 version corrected only a misspelled word.
- Kolmogorov's 0–1 law • Corrected 2025-11-14 after the lecture
- Conditional expectation • New 2025-11-13
Lectures
Tuesday 10:15-12:00, GL-RFB R4
Friday 8:15-10:00, GL-RFB R4
Exercises (Øvinger) Friday 14:15-15:00, GL-OP B3
Lectures will be given in English.
Lectures will start on 19 August. There will be no exercises the first week.
Lectures week 34
The lectures in week 34 will be given by Magnus B. Landstad
* Office: room 1129, Sentralbygg 2
* Email: magnus.landstad@ntnu.no
I suggest that you take a look at what you know about Riemann integration. Look in your calculus books or at the beginning of https://measure.axler.net/SupplementMIRA.pdf
Also read the Preface for Students in the book and I will read the the Preface for Instructors.
"Referansegruppe"
Antonije Mirkovic antonijm [at] stud [dot] ntnu [dot] no, Jacek Stankiewicz jaceks [at] stud [dot] ntnu [dot] no, Jørgen Sønstabø jorgerso [at] stud [dot] ntnu [dot] no.
Lecture plan
| Week | Topic | Exercises |
|---|---|---|
| 34 | Ch. 1 (Textbook): Mostly without proofs. Suppment: Sec. A: Parts without proofs. Sec. C: Parts, only for \(\mathbb{R}\). Sec. D: All, except 0.56-0.57. Sec. E: Should be known for \(\mathbb{R}\). | |
| 35 | Ch. 2: 2.1–2.38. | Suppl: C: 2, 5, 6, 10, 13, 14. D: 8, 9, 13, 15, 16. E: 6. Ch. 1A: 1, 4, 13. Ch. 1B: 1, 5. Ch. 2A: 1, 2. |
| 36 | Ch. 2: 2.39-2.66. (no proof for 2.65) | Ch. 1A: 13. Ch. 2A: 11. Ch. 2B: 1, 2, 11. |
| 37 | Ch. 2: 2.67-2.84. | Ch. 2A: 7, 9. Ch. 2B: 6,11. Ch. 2C: 9, 11. |
| 38 | Ch. 2: 2.85 – end of chapter 2. | Ch. 2D: 5, 6, 10, 12. Ch. 2E: 5, 12. • solutions |
| 39 | Tue: Ch. 3: 3.1–3.18 (I followed Notes on the Lebesgue integral. Fri: Ch. 3: 3.19-31. | Ch. 3A: 1, 2, 3, 5, 6, 9. |
| 40 | Tue: Ch. 3: 3.34–3.48. Ch. 4: 4.1-4.5. Fri: Luzin and \(L^1\) approximation combined and Ch. 4: 4.6-4.10. | Ch. 3A: 17. Ch. 3B: 3, 5, 12. |
| 41 | Tue: Ch. 4: 4.16–4.24. Ch. 5: 5.1–5.8. Fri: Ch. 5: 5.9–5.27. | Ch. 3B: 10, 12, 16a. Ch. 4A: 1, 2, 6. |
| 42 | Tue: Ch. 5: 5.28–5.36. Fri: Ch. 5: 5.37–, replacing 5.41–5.45 by this: A uniqueness theorem and its consequences up to Corollary 5, which was done in a very quick, hand-wavy fashion. | Ch. 4B: 3, 5, 6. Ch. 5A: 2, 9, 10. |
| 43 | Tue: Started with Cor. 5, and completed the notes. Ch. 7: 7.1–7.8. Fri: Ch. 7: 7.9–7.20. We did not complete the proof of 7.20. | Ch. 5B: 1. Ch. 5C: 8, 12, 14. |
| 44 | Tue: Ch. 7: 7.20-7.25. Ch. 9: 9.1-9.7. Fri: Ch. 9: 9.23-9.40. | Ch. 6A: 14. Ch. 7A: 1,2,7,8. Ch. 7B: 8. |
| 45 | Tue: Ch. 9: 9.41-9.42.Ch. 12: 12.1-12-7. Fri: Ch. 12: 12.8-12.22. | Ch. 7A: 4, 5, 17, 18, 19. Ch. 7B: 13. |
| 46 | Tue: Ch. 12: 12.23-12.31. Fri: 12.32–12.33 (done differently), Kolmogorov's zero-one law (see notes above) | Ch. 12: 3, 4, 10, 15. |
| 47 | Tue: Ch. 12: 12.35-12.38. Strong law of large numbers. Fri: Exam TMA4225 from 2023, except Problem 2 and 5c. | Q&A ("Spørretime") |
Remarks
Littlewood's three principles:
- A Lebesgue measurable set with finite measure is almost a union of finitely many open intervals (see Ch. 2D exercise 6)
- A Lebesgue measurable function is almost continuous (Luzin's theorem)
- A pointwise convergent sequence of functions is almost uniformly convergent (Egoroff's theorem)
Curriculum
Curriculum (all from Axler): Chapters 1-5, 7, 9 (pp. 256-259, 267-277), 12. Plus notes posted on this web page.
Exam
Written exam (in English, but answers may be written in English or a Scandinavian language). Date: 9 December 2025 at 15:00. Time: 4 hours. Venue: See here 3 days before the exam date.
Permitted aids: (Support material code D) No printed or hand-written support material is allowed. A specific basic calculator is allowed.