Prof. Dr. Francesca Da Lio

Analysis III (Masstheorie), D-MATH

Fall Semester 2023

Lecturer: Prof. Francesca Da Lio

Exercise hours coordinator: Gerard Orriols Gimenez

Diary of the lectures

#Week Date Content Notes Reference
1 20/22.09.2022 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 27/29.09.2023 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). Class Notes Section 1.2.1 & 1.2.2 in the Lecture Notes
3 4-6.10.2023 Definition of a covering. Proof of Theorem 1.2.18 (construction of a measure). Definition of a pre-measure. Examples. Carathéodory-Hahn extension. Proof of Theorem 1.2.20. Statement 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 Lebesqgue measure is a Borel measure. Regularity properties of Lebesgue measure. Proof of Theorem 1.3.7 and Theorem 1.3.8. Sufficient and necessary conditions for the Lebesgue measurability. Comparison between Jordan and Lebesgue measures. Statement of Theorem 1.4.1 Class Notes Section 1.3 and 1.4. in the Lecture Notes.
4 11-13.10.2023 Proof of Theorem 4.1. Example of a Jordan non measurable set. Example of a Lebesgue non measurable set : Vitali set. Dyadic decomposition of R^n. Proof of the fact that Lebesgue measure is Borel and Borel regular. Class Notes 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,3) The Axiom of Choice and its implications in mathematics
5 18-20.10.2023 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.Lebesgue-Stieltjes measures. Definition of metric measures in R^n. Proof of Theorem 1.7.4 and Theorem 1.7.5. Definition of the s-Hausdorff measure of step delta. Class Notes Sections 1.6, 1.7 & 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
6 25-27.10.2023 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. Radon Measures (only definition and statement of some properties) Class Notes Section 1.8 and 1.9 in Lecture Notes
7 1-3.11.2023 Measurable functions (definition and properties). sup and inf of measuarble functions. Approximation of nonnegative measurable functions by simple functions. Littlewood's three principles. Proof of Egoroff's Theorem. Some counter-examples. Class Notes Sections 2.2 & 2.3 in the Lecture Notes
8 8-10.11.2023 Statement of Lusin's theorem. Counter-example to the case $\epsilon=0$. Convergence in measure. 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 Sections 2.4 & Section 3.1 in the Lecture Notes
9 15-17.11.2023 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,Statement of Lemma 3.1.16, Proof of Corollary 3.1.17 and Proposition 3.1.18. Comparison between Riemann and Lebesgue Integrals. Convergence results. Statement of Fatou's Lemma. Class Notes Sections 3.1 & 3.2 in the Lecture Notes
10 22-24.11.2023 Proof of Fatou's Lemma, Proof of Beppo Levi Theorem, Proof of Dominated Convergence Theorem, 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 Sections 3.3-3.6 in the Lecture Notes
11 29.11-1.12.2023 Comments on Vitali Theorem and its link with Lebesgue Theorem. Exercise 3.6.7. L^p spaces, Young Inequality, Hölder Inequality, Minkowski Inequality Class Notes Section 3.7 in Lecture Notes until page 108.
12 6-8.12.2023 Completeness of L^p spaces (Proof of Theorem 3.7.13). Product of measures. Fubini and Tonelli theorems (only statements). Examples and applications Class Notes Section 3.7 in Lecture Notes until page 112, Section 4.1.
13 13-15.12.2023 Solutions of two esercises in Section 4.2. Change of Variable Formula and Applications. Definition of the convolution. Proof ofLemma 4.4.1, Theorem 4.4.5, Corollary 4.4.6 Class Notes Section 4.3 & 4.4. until page 143.
14 20-22.12.2023 Proof of Proposition 4.4.8. Proof of Theorem 3.7.15 and remarks on the case $p=+\infty$ Class Notes Section 4.4 & 3.7
**** **** Frohe Weihnachten und Viel Erfolg! ****

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