Analysis III, D-MATH Fall Semester 2022
Lecturer: Prof. Francesca Da Lio
Exercise hours coordinator: Gerard Orriols Gimenez
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Lecture Notes (These notes will be continuously upadated during the course)
Diary of the lectures
# | Date | Content | Notes | Reference |
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1 | 21.09.2022 | Slides of presentation of the course, Preliminary notations and definitions | Class Notes | Section 1.1.1 in the Lecture Notes (Page 1) & Class Notes. For curiosity: A proof of De Morgan Identities |
2 | 23.09.2022 | limsup, liminf of sequences of sets, limit of monotone sequences of sets, algebras, sigma-algebras, definition and examples. | Class Notes | Section 1.1.2 in the Lecture Notes |
3 | 28.09.2022 | 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. | Class Notes | Section 1.2.1 in the Lecture Notes |
4 | 30.09.2022 | 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 1.2.17 (construction of a measure). | Class Notes | Section 1.2.2 in the Lecture Notes |
5 | 5.10.2022 | Definition of a pre-measure. Examples. Carathéodory-Hahn extension. Proof of Theorem 1.2.20. Statement of Theorem 1.2.21 Uniqueness Carathéodory-Hahn extension. | Class Notes | Section 1.2 until page 21 in the Lecture Notes. |
6 | 7.10.2022 | Definition of a multi-interval, volume of a multi-interval. Sigma-subadditivity of the volume. Dyadic decomposition of the Euclidean space.Definition of Lebesqgue measure. The Lebesqgue measure is a Borel measure. Regularity properties of Lebesgue measure. Proof of Theorem 1.3.7 | Class Notes | Section 1.3 until page 26 in the Lecture Notes. |
7 | 12.10.2022 | Sufficient and necessary conditions for the Lebesgue measurability. Comparison between Jordan and Lebesgue measures. Theorem 1.4.1 | Class Notes | Section 1.3 and Section 1.4 in the Lecture Notes. |
8 | 14.10.2022 | Examples of Jordan nonmeasurable sets. The Lebesgue measure is a Borel regular measure (Corollary 1.4.4). Vitali set. | Class Notes | Section 1.4 & 1.5. Section 1.4 & Section 1.5 in the Lecture Notes. For curiosity: 1) An example of Lebesgue measurable set in R which is not Borel,2) Banach-Tarski theorem,3) Some pathological sets in the standard theory of Lebesgue measure (Bachelor thesis). Some references on the axiom of the choice: 1) A look at the world without the axiom of the choice, 2) A Model of Set-Theory in Which Every Set of Reals is Lebesgue Measurable |
9 | 19.10.2022 | Every countable set in R has measure zero. Description of the Cantor triadic set : example of an uncountable with zero Lebesgue measure. Its representation in base b=3 | Class Notes | Section 1.5 & 1.6 in the Lecture Notes. |
10 | 21.10.2022 | 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 | Class Notes | Section 1.8 until page 47 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 |
11 | 26.10.2022 | Definition of the Hausdorff dimension of a set of R^n. Some remarks about the Hausdorff dimension of subsets of R^n, description of Cantor dust | Class Notes | Section 1.8 in the Lecture Notes. |
12 | 28.10.2022 | Dimension of Cantor dust. Radon Measures. Measurable functions (definition and properties) | Class Notes | Section 1.9, Section 2.1 in the Lecture Notes. |
13 | 2.11.2022 | sup and inf of measarble functions. Approximation of nonnegative measurable functions by simple functions. | Class Notes | Section 2.2 & 2.3 until page 61 in the Lecture Notes |
14 | 4.11.2022 | Littlewood's three principles. Proof of Egoroff's Theorem and Lusin 's Theorem. Some counter-examples | Class Notes | Section 2.3 until page 65 in the Lecture Notes |
15 | 9.11.2022 | Proof of Lusin 's Theorem (continued). Convergence in measure | Class Notes | Section 2.3 until page 67 in the Lecture Notes |
16 | 11.11.2022 | Relation between convergence in measure and the almost everywhere convergence. Definition of the integral with respect to a given Radon measure on R^n. Proof of Propositions 3.1.7 | Class Notes | Section 2.3 & 2.4 & 3.1 until page 74 |
17 | 16.11.2022 | Proof of Propositions 3.1.7 continued. Proof of Proposition 3.1.10 | Class Notes | Section 2.3 & 2.4 & 3.1 until page 75 |
18 | 18.11.2022 | Linearity of the integral (Proof of Theorem 3.1.15). Comparison between Riemann and Lebesgue Integrals. Convergence results. Fatou's Lemma. Example 3.3.2. | Class Notes | Section 3.2 & 3.3. FOR CURIOSITY: On the behavior at infinity of an integrable function, by Emmanuel Chaissaigne |
19 | 23.11.2022 | Fatou's Lemma (continued), Beppo Levi's Theorem, Dominated Convergence Theorem | Class Notes | Section 3.3 |
20 | 25.11.2022 | Two Applications of Monotone Convergence Theorem (integral of a series of functions and Borel-Cantelli Lemma), Absolute Continuity of Integrals, Vitali's Theorem | Class Notes | Section 3.4 & 3.5 until page 96. For the applications see class notes. |
21 | 30.11.2022 | Link between Lebesgue summability and Riemann integrability in the generalized sense (see Exercise 3.6.7), proof that sin(x)/x is not Lebesgue summable but only Riemann integrable in the generalized sense. Intruduction of L^p spaces. Examples. | Class Notes | Section 3.6 & 3.7 until page 102 |
22 | 2.12.2022 | Young Inequality, Hölder Inequality,Minkoswki Inequality, L^p is a normed space. Proof that L^{\infty} is complete. | Class Notes | Section 3.7 until page 108 |
23 | 7.12.2022 | Prood of the fact that L^p is a complete space for 1\le p<+\infty. Definition of product measures | Class Notes | Section 3.7 & Section 4.1 until page 118 |
24 | 9.12.2022 | Fubini and Tonelli Theorems (only statements) and applications (see Class Notes) | Class Notes | Section 4.2 until page 132. |
25 | 14.12.2022 | Change of Variable Formula and Application. Definition of the convolution. | Class Notes | Sections 4.3 and 4.4 until page 138. |
26 | 16.12.2022 | Proof of Theorem 4.4.5, Corollary 4.4.6, Theorem 4.4.8 | Class Notes | Sections 4.4 |
27 | 21.12.2022 | Conclusion Proof of Theorem 4.4.8. Exercises for preparation written exam | Class Notes | Sections 4.4 |
28 | 23.12.2022 | Exercises for preparation written exam | Frohe Weihnachten | Viel Erfolg! |
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.
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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)