Please email Henk Bruin for more information on this course.

The proseminar will be given by Davide Ravotti.

Information from the Course catalogue and for the Proseminar.

No Exercise Class in the first week (October 7).

Day | Time | Room | Type | From | To |
---|---|---|---|---|---|

Monday | 8:00-9:30 | HS2 | Lecture | 3.10.2022 | 30.1.2023 |

Tuesday | 8:00-9:30 | HS2 | Lecture | 4.10.2022 | 31.1.2023 |

Friday | 8:00-8:45 | SR10 | Exercises | 14.10.2022 | 27.1.2023 |

Week 1 | Type | Topic/Material |
---|---|---|

October 3 | Lecture: Bruin | Introduction, Sigma-Algebras |

October 4 | Lecture: Bruin | |

October 7 | Exercises: Ravotti | No exercise class |

Week 2 | Type | Topic/Material |

October 10 | Lecture: Ravotti | Measures and outer measures, measurable sets Bass Chapter 3 and 4.1 |

October 11 | Lecture: Ravotti | |

October 14 | Exercises: Ravotti | Bass, Chapter 2, Exercises 1,2,3,5,7,9 |

Week 3 | Type | Topic/Material |

October 17 | Lecture: Bruin | Lebesgue-Stieltjes measure and outer measure Caratheodory extension theorem, Bass Chapter 4 |

October 18 | Lecture: Bruin | |

October 21 | Exercises: Ravotti | Bass, Chapter 3, Exercises 2-7 and 9. |

Week 4 | Type | Topic/Material |

October 24 | Lecture: Ravotti | Measurable functions Bass Chapter 5 |

October 25 | Lecture: Ravotti | |

October 28 | Exercises moved to Monday 31 | |

Week 5 | Type | Topic/Material |

October 31 | Exercises: Bruin 8:45-9:30 | Chapter 3: 4,5,6,7 and Chapter 4: 1,2 |

November 1 | No class | All Saints Day |

November 4 | Exercises: Ravotti | Bass, Chapter 4, Exercises 4.6, 47, Chapter 5, 5.4, 5.5 |

Week 6 | Type | Topic/Material |

November 7 | Lecture: Bruin | Lebesgue Density Theorem |

November 8 | Lecture: Bruin | Legesbue Integral Bass Chapter 6 |

November 11 | Exercises: Ravotti | Bass, Chapter 5, Exercises 5.1, 5.3, Chapter 6, 6.1 and 6.4 |

Week 7 | Type | Topic/Material |

November 14 | Lecture: Ravotti |
Limit Theorems Bass, Chapter 7 |

November 15 | Lecture: Ravotti | |

November 18 | Exercises: Ravotti | Exercises: 6.5 + three extra on the Lebesgue density theorem |

Week 8 | Type | Topic/Material |

November 21 | Lecture: Bruin |
Properties for f = 0 Riemann integral |

November 22 | Lecture: Bruin | Chapter 8 and 9 |

November 25 | Exercises: Ravotti | Exercises: 7.5, 7.10, 7.12, 7.17 |

Week 9 | Type | Topic/Material |

November 28 | Lecture: Ravotti |
Types of Convergence, Product measures Chapter 10 + 11 |

November 29 | Lecture: Ravotti | |

December 2 | Exercises: Ravotti | Exercises: 7.27, 8.2, 8.7, 8.8 |

Week 10 | Type | Topic/Material |

December 5 | Lecture: Ravotti |
Fubini's Theorem, Signed measures Chapter 11 + 12 |

December 6 | Lecture: Bruin | |

December 9 | Exercises: Ravotti | Exercises: 10.2, 10.4, 10.9, 11.4 |

Week 11 | Type | Topic/Material |

December 12 | Lecture: Bruin |
Jordan decomposition, absolute continuity, Radon-Nikodym derivative Chapter 12 + 13 |

December 13 | Lecture: Ravotti | |

December 16 | Exercises: Ravotti | |

Week 12 | Type | Topic/Material |

January 9 | Lecture:Ravotti | L^{p}-spaces, Hölder and Minkovski inequality, Completions |

January 10 | Lecture:Ravotti | |

January 13 | Exercises: Ravotti | |

Week 13 | Type | Topic/Material |

January 16 | Lecture: Bruin | Convolutions, bounded functional, Chapter 15 |

January 17 | Lecture: Bruin | |

January 20 | Exercises: Bruin |
Exercises 15.1, 15.6, 15.7 and the following:
(a) Let f be the function on R^{n} defined by f(x) = e^{-1/(1-|x|^2)} for |x|<1 and f(x) = 0 otherwise. Prove that f is infinitely differentiable on R^{n}. Is it real analytic? |

Week 14 | Type | Topic/Material |

January 23 | Lecture:Bruin | Riesz Representation Lemma, Chapter 17 |

January 24 | Lecture:Bruin | |

January 26 | Exercises: Ravotti |

By oral examination on appointment.

- Richard F. Bass, Real Analysis for Graduate Students, Version 4.3.

This text, of which we will roughly cover Chapters 1-13, can be downloaded for free. - Walter Rudin, The principles of real analysis, (International Series in Pure and Applied Mathematics) 3rd Edition, ISBN-10 007054235X (Chapter 6). Also online
- Note on the Lebesgue Density Theorem
- The notes of Roland Zweimüller can be found on Moodle.
- Other set of notes can be found on Moodle.

Updated August 2022