Prof. Dr. Francesca Da Lio

Analysis III, D-MATH Fall Semester 2022

Lecturer: Prof. Francesca Da Lio

Exercise hours coordinator: Gerard Orriols Gimenez

Diary of the lectures

# Date Content Notes Reference
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!

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