Please email Henk Bruin for more information on this course.
Information from the Course catalogue and for the Proseminar.
| Day | Time | Room | Type | From | To |
|---|---|---|---|---|---|
| Wednesday | 8:00 - 9:30 | HS11 | Lecture | 01.10.2025 | 28.01.2026 |
| Thursday | 11:30 - 13:00 | SR7 | Proseminar | 02.10.2025 | 29.01.2026 |
| Friday | 13:15 - 14:45 | HS13 | Lectures | 03.10.2025 | 30.01.2026 |
| Week 1 | Type | Topic/Material | |
|---|---|---|---|
| October 1 | Lecture | Introduction, Sigma-Algebras Bass 5.0 Chapter 2 | |
| October 2 | Proseminar Not this week yet | Exercises for the Proseminar of Week 2: Exercise Bass 5.0: Exercise 2.1 - 2.9 and 20.2, 20.7, 20.12 In the attached file, I collected chatGPT solutions for five of these. If you tick these, you must be able to discuss the chatGPT solution. Are they correct? If not, can you correct them? You should also be able to clarify the difficult or possibly obscure steps. I will create a Moodle page with tick-list shortly. Deadline for this tick-list is Thursday October 9 at 10am | |
| October 3 | Lecture | Introduction to topology Sets of 1st and 2nd category | |
| Week 2 | Type | Topic/Material | |
| October 8 | Lecture | outer measure. Carathéodory's Theorem
Bass 5.0 Chapter 4.1 | |
| October 9 | Proseminar | Exercises for the Proseminar of Week 2: Exercise Bass 5.0: Exercise 2.1 - 2.9 and 20.2, 20.7, 20.12 In the attached file, I collected chatGPT solutions for five of these. If you tick these, you must be able to discuss the chatGPT solution. Are they correct? If not, can you correct them? You should also be able to clarify the difficult or possibly obscure steps. Deadline for this tick-list is Thursday October 16 at 10am | |
| October 10 | Lecture | Lebesgue measure Non-measurable sets Bass 5.0 Chapter 4.2-4.4 Oxtoby Chapter 5 | |
| Week 3 | Type | Topic/Material | |
| October 15 | Lecture | Non-meaurable sets. Set swith the Baire property.
Bass 5.0 Chapter 4.4 Oxtoby Chapter 4 | |
| October 16 | Proseminar | Exercises for the Proseminar of Week 3: Exercise Bass 5.0: Exercise 3.2 - 3.10 Deadline for the check-list is Thursday October 16 at 10am | |
| October 17 | Lecture | No lecture this Friday | |
| Week 4 | Type | Topic/Material | |
| October 22 | Lecture | Non-meaurable sets. Baire spaces Theorem of Baire Oxtoby Chapter 9 | |
| October 23 | Proseminar | Exercises for the Proseminar of Week 4: Bass 5.0: Exercise 4.1, 4.3-4.10
Deadline for the check-list is Thursday October 23 at 10am | |
| October 24 | Lecture | Carathéory Extension Theorem Lebesgue density theorem. Bass 5.0 Chapter 4.5 Lebesgue Density Theorem | |
| Week 5 | Type | Topic/Material | |
| October 29 | Lecture | Proof of Lebesgue Density Theorem Integration Lebesgue Density Theorem Bass 5.0 Chapter 5.1 - 5.2 | |
| October 30 | Proseminar | Exercises for the Proseminar of Week 5:
Bass 5.0: Exercise 4.11, 4.12, 4.14
Deadline for the check-list is Thursday October 30 at 10am | |
| October 31 | Lecture | Measurable functions Lusin's Theorem Bass 5.0 Chapter 5.3 | |
| Week 6 | Type | Topic/Material | |
| November 5 | Lecture |
Lusin's Theorem Integrable functions Monotone Convergence Theorem Bass 5.0 Chapter 6 and 7 | |
| November 6 | Proseminar | Exercises for the Proseminar of Week 6:
Bass 5.0: Exercises 5.1-5.10.
Deadline for the check-list is Thursday November 6 at 10am | |
| November 7 | Lecture | Fatou's Lemma Dominated Convergence Theorem Lebesgue vs. Riemann integrable Bass 5.0 Chapter 7 and 9 | |
| Week 7 | Type | Topic/Material | |
| November 12 | Lecture |
Lebesgue vs. Riemann integrable Diophantine and Liouville numbers Bass 5.0 Chapter 9 Oxtoby Chapter 2 | |
| November 13 | Proseminar | Exercises for the Proseminar of Week 7:
Bass 5.0: Exercises 6.3, 6.5, 6.8 and 6.9.
Deadline for the check-list is Thursday November 13 at 10am | |
| November 14 | Lecture |
Diophantine and Liouville numbers Product measures Oxtoby Chapter 2 Bass 5.0 Chapter 11.1 | |
| Week 8 | Type | Topic/Material | |
| November 19 | Lecture |
Fubini's Theorem Bass 5.0 Chapter 11.2 | |
| November 20 | Proseminar | Exercises for the Proseminar of Week 8:
Bass 5.0: Exercises 7.2-7.7 and 7.13. (If you really have trouble with 7.2, try this video.) Deadline for the check-list is Thursday November 20 at 10am | |
| November 21 | Lecture |
signed measures, Hahn and Jordan decompositions
Bass 5.0 Chapter 12 | |
| Week 9 | Type | Topic/Material | |
| November 26 | Lecture |
We start today at 8:45 - from your side, read the proofs of
the Hahn decomposition and the Jordan decomposition
Hahn and Jordan decompositions The Radon-Nikodym Theorem Bass 5.0 Chapter 12+13 | |
| November 27 | Proseminar | Exercises for the Proseminar of Week 9:
Bass 5.0: Exercises 11.4, 11.5, 11.6, 11.9, 11.14. Deadline for the check-list is Thursday November 27 at 10am | |
| November 28 | Lecture |
The Radon-Nikodym Theorem
Bass 5.0 Chapter 12 | |
| Week 10 | Type | Topic/Material | |
| December 3 | Lecture |
Lp-space, Hölder and Minkovski inequalities Bass 5.0 Chapter 15.1-2 | |
| December 4 | Proseminar | Exercises for the Proseminar of Week 10:
Bass 5.0: Exercises 12.1, 12.3, 12.5, 12.7, 13.1, 13.5, 13.9, 13.12
Deadline for the check-list is Thursday December 4 at 10am | |
| December 5 | Lecture |
Convolutions Bass 5.0 Chapter 15.3 | |
| Week 11 | Type | Topic/Material | |
| December 10 | Lecture |
Linear functionals and dual spaces Bass 5.0 Chapter 15.4 | |
| December 11 | Proseminar | Exercises for the Proseminar of Week 11:
Bass 5.0: Exercises 15.1-15.5, 15.8, 15.11, 15.12 Deadline for the check-list is Thursday December 11 at 10am | |
| December 12 | Lecture |
Duality proof of Lp
Partitions of unity Bass 5.0 Chapter 15.4 and 17.1 | |
| Week 12 | Type | Topic/Material | |
| December 17 | Lecture |
Riesz Representation Theorem Bass 5.0 Chapter 17.2 | |
| December 18 | Proseminar | Exercises for the Proseminar of Week 12:
Bass 5.0: Exercises 15.15-18, 15.24, 15.29 Deadline for the check-list is Thursday December 18 at 10am | |
| December 19 | No Lecture | ||
| Week 13 | Type | Topic/Material | |
| January 7 | Lecture |
Lecture cancelled. Bass 5.0 Chapter | |
| January 8 | Proseminar | Proseminar cancelled | |
| January 9 | No Lecture | ||
| Week 14 | Type | Topic/Material | |
| January 14 | Lecture |
Regular measures Summary of the Measure Theory so far. | |
| January 15 | Proseminar | Exercises for the Proseminar of Week 14: Bass 5.0: Exercises 15.25, 15.26, 17.1, 17.3, 17.4, 17.5, 17.8, 17.9||
| January 16 | Lecture |
Fractals and Hausforff dimension/measures Troscheit's Notes Chapter 1 | |
| Week 15 | Type | Topic/Material | |
| January 21 | Lecture |
| |
| January 22 | Proseminar | Exercises for the Proseminar of Week 15 on this exercise sheet.||
| January 23 | Lecture | ||
| Week 16 | Type | Topic/Material | |
| January 28 | Lecture |
| |
| January 29 | Proseminar | Exercises for the Proseminar of Week 16||
| January 23 | Lecture |
By oral examination by appointment.