Analysis III (Masstheorie), D-MATH
Fall Semester 2024
Lecturer: Prof. Francesca Da Lio
Exercise hours coordinator: Antonio Marini
- Lecture Notes (These notes will be continuously upadated during the course)
- Class Notes
- Course Webpage
Diary of the lectures
| #Week | Date | Content | Notes | Reference |
|---|---|---|---|---|
| 1 | 18/20.09.2024 | Slides of presentation of the course. Preliminary notations and definitions, limsup, liminf of sequences of sets, limit of monotone sequences of sets, rings, algebras, sigma-algebras, examples. | Class Notes | Sections 1.1.1 and 1.1.2 in the Lecture Notes. For curiosity: A proof of De Morgan Identities; A Non-Borel set |
| 2 | 25/27.09.2024 | Sigma-algebra of Borel sets, examples, additive and sigma-additive functions, proof of the fact that an additive function is sigma-additive iff it is subadditive. Definition of a measure and of measurable sets. Proof of Theorem 1.2.10 (the set of measurable sets is a sigma algebra), definition of a measure Space, Exercise 1.2.12., Proof of Theorem 1.2.13 (continuity properties of a measure). Definition of a covering. Proof of Theorem 1.2.18 (construction of a measure). Definition of a pre-measure. Examples. Carathéodory-Hahn extension. Statement of Theorem 1.2.20. | Class Notes | Section 1.2.1 & 1.2.2 in the Lecture Notes |
| 3 | 2/4.09.2024 | Proof of Theorem 1.2.21. about Uniqueness Carathéodory-Hahn extension. Definition of a multi-interval, volume of a multi-interval. Sigma-subadditivity of the volume. Definition of Lebesqgue measure. The Lebesgue measure is a Borel measure. Regularity properties of Lebesgue measure. Statement of Theorem 1.3.7 and Theorem 1.3.8. Sufficient and necessary conditions for the Lebesgue measurability. | Class Notes | Section 1.3 and 1.4. in the Lecture Notes. |
| 4 | 9/11.10.2024 | Proof of Theorem 1.3.7 and Theorem 1.3.8. Sufficient and necessary conditions for the Lebesgue measurability. Proof of the fact that Lebesgue measure is Borel regular (Corollary 1.4.4). Comparison between Jordan and Lebesgue measures. Proof of Theorem 1.4.1. Examples of non Jordan measurable sets. Example of a non Jordanmeasurable set : Vitali set. | Class Notes | Section 1.3 and 1.4. in the Lecture Notes. For curiosity: 1) An Overview of Jordan Measure (Bachelor Thesis). 2) An example of Lebesgue measurable set in R which is not Borel,3) Banach-Tarski theorem,4) Some pathological sets in the standard theory of Lebesgue measure (Bachelor thesis). Some references on the axiom of the choice: 5) A look at the world without the axiom of the choice, 6) A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable,7) The Axiom of Choice and its implications in mathematics |
| 5 | 16-18.10.2024 | Description of the Cantor triadic set : example of an uncountable with zero Lebesgue measure. Its representation in base b=3.Lebesgue-Stieltjes measures. Definition of metric measures in $R^n$. Proof of Theorem 1.7.4 and Theorem 1.7.5. | Class Notes | Sections 1.6, 1.7 in the Lecture Notes |
| 6 | 23-25.10.2024 | Introduction of Hausdorff measure. Proof of the fact that the zero dimensional Hausdorff measure is the counting measure. Definition of metric measure. Proof of Theorem 1.8.3 (the s-dimensional Hausdorff measure is a Borel regular measure). Proof of Lemma 1.8.5, Example 1.8.6. Definition of the Hausdorff dimension of a set of $R^n$. Some remarks about the Hausdorff dimension of subsets of $R^n$. Examples of fractals. | Class Notes | Sections 1.8 in the Lecture Notes. Just for curiosity: 1) Fractals and 2) Prof. R. Mingione's talk about fractals (in italian) and 3) The unlimited s- Hausdorff measure is not in general a Borel measure |
| 7 | 1.11.2024 | Radon Measures (only definition and statement of some properties).Measurable functions (definition and properties). sup and inf of measuarble functions. Approximation of nonnegative measurable functions by simple functions. | Class Notes | Sections 1.9, 2.1, 2.2 in the Lecture Notes |
| 8 | 6-8.11.2024 | Littlewood's three principles. Proof of Theorem 2.3.1 (Egoroff's Theorem). Some counter-examples. Proof of Theorem 2.3.3 (Lusin's Theorem) Counter-example to the case $\epsilon=0$. Convergence in measure. Relation between convergence in measure and the almost everywhere convergence. | Class Notes | Sections 2.3 & 2.4 in the Lecture Notes |
| 9 | 13-15.11.2024 | Definition of the integral with respect to a given Radon measure on R^n. Proof of Propositions 3.1.7. Proof of Proposition 3.1.6, 3.1.7, 3.1.9, 3.1.10, Corollary 3.1.11 and Corollary 3.1.13. Statement of Theorem 3.1.14 (linearity of integral), Tchebychev Inequality. Comparison between Riemann and Lebesgue Integrals. | Class Notes | Sections 3.1 & 3.2 in the Lecture Notes |
Recommended bibliography (Undergraduate-Master level):
- Lawrence Evans and Ronald Gariepy, Measure Theory and Fine Properties of Functions, Textbooks in Mathematics, CRC Press, 2015.
- Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications
- Michael Struwe, Analysis III: Mass und Integral, Lecture Notes, ETH Zürich, 2013.
- Piermarco Cannarsa and Teresa D'Aprile, Lecture Notes on Measure Theory and Functional Analysis, Lecture Notes, University of Rome, 2006.
- Terence Tao, An Introduction to Measure Theory, American Mathematical Society, 2011.
- Herbert Amann and Joachim Escher, Analysis III, 2009 Birkhäuser Verlag AG.
-
Recap of basic topology notions: Chapter 4 in the lecture notes Analysis I and II Michael Struwe
Further reading:
- W.F. Eberlein, Notes on Integration I: The Underlying Convergence Theorem, Comm. Pure Appl. Math. 10 (1957), 357–360.
- Ask yourself dumb questions – and answer them! (by Terence Tao);
- How to write Mathematics (by Paul R. Halmos)