Prof. Dr. Francesca Da Lio

Analysis III (Masstheorie), D-MATH

Fall Semester 2025

Lecturer: Prof. Francesca Da Lio

Exercise hours coordinator: Antonio Marini

Course Description

This course introduces the modern framework of measure and integration, focusing on the Lebesgue measure and integral on R^n. Students learn how this generalizes Riemann integration, provides powerful convergence theorems, and forms the foundation for modern analysis, probability, and functional analysis.

Learning Outcomes

By the end of the course, students will be able to:

1 Explain the motivation for measure theory and how it extends classical notions of length, area, and integration.
2   Define σ-algebras, measurable sets, measures, and measurable functions, and give concrete examples.
3   Construct the Lebesgue measure on R^n and compute simple examples.
4   Develop the Lebesgue integral for simple and general functions.
5   Apply the key convergence theorems: Monotone Convergence, Fatou’s Lemma, and Dominated Convergence.
6   Understand the relationship between Lebesgue and Riemann integration.
7   Use product measures and apply Fubini’s Theorem to compute double integrals.
8   Explore ( L^p ) spaces and their properties, including inequalities and completeness.
 9  Write clear, rigorous proofs and explanations involving measurable sets, functions, and integrals.

Diary of the lectures

#Week Date Content Notes Reference
1 17-19.09.2025 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. Sigma-algebra of Borel sets, examples, definition of additive and sigma-additive functions Class Notes Sections 1.1.1 and 1.1.2 in the Lecture Notes. For curiosity:1) A proof of De Morgan Identities; 2)A Non-Borel set; 3) The Axiom of Choice and its implications in mathematics
2 24-26.09.2025 Remark 1.2.2 (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 1-4.10.2025 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. 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 Lebesgue measure is a Borel measure. Regularity properties of Lebesgue measure. Statement of Theorem 1.3.7. a Class Notes Section 1.3 and 1.4. in the Lecture Notes.
4 8-10.10.2025 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). Proof of Lemma 1.3.4.Comparison between Jordan and Lebesgue measures. Proof of Theorem 1.4.1. Examples of non Jordan measurable sets. Class Notes Section 1.3 and 1.4. in the Lecture Notes.
5 15-17.10.2025 Example of a non Lebesgue measurable set: Vitali set. Description of the Cantor triadic set : example of an uncountable with zero Lebesgue measure. Its representation in base b=3.Lebesgue-Stieltjes measures. Class Notes Sections 1.6, 1.7 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
6 22-24.10.2025 Proof of the fact that Lebesgue measure is Borel regular (Corollary 1.4.4). Definition of metric measures in $R^n$. Proof of Theorem 1.7.4 and Theorem 1.7.5. Introduction of Hausdorff measure. Proof of the fact that the zero dimensional Hausdorff measure is the counting measure. Definition of metric measure. Class Notes Sections 1.7 & 1.8 in the Lecture Notes.
7 29-31-10-2025 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
8 5-7.11.2025 Radon Measures (only definition and statement of some properties).Measurable functions (definition and properties). sup and inf of measuarble functions. Proof of Theorem 2.2.6.( Approximation of nonnegative measurable functions by simple functions). Remark 2.2.8. Proof of Theorem 2.3.1 (Egoroff's Theorem). Exercise 2.3.2 Counter-example to the case $\delta=0$. Class Notes Sections 1.9, 2.1, 2.2, 2.3 & 2.4 in the Lecture Notes. Littlewood's three principles (notes written by Maran Mohanarangan).
9 12-14.11.2024 Littlewood's three principles. Proof of Theorem 2.3.1 (Egoroff's Theorem). Some counter-examples. Statemnt of Theorem 2.3.3 (Lusin's Theorem) Counter-example of Lusin Theorem 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. Class Notes Sections 2.3 & 2.4 & 3.1 in the Lecture Notes

Recommended bibliography (Undergraduate-Master level):