Prof. Dr. Francesca Da Lio

Mass und Integral, D-MATH Spring Semester 2020

Lecturer: Prof. Francesca Da Lio

Coordinator: Jerome Wettstein

Class Content

Lecture Topic Bibliograpghy
19.02.2020 Presentation of the course. Algebras, sigma-algebras, All Section 1.1 of Lecture Notes
21.02.2020 Examples of sigma-algebras (see Ex 1.1.6). Additive and sigma-additive functions, measures. Carathéodory criterium of measurability. Proof of Theorems 1.2.10, Section 1.2.1, (until page 10)
26.02.2020 End of Proof of Theorem 1.2.10, Proof of Theorem 1.2.13. Definition of a covering. Construction of a measure. Proof of Theorem 1.2.17 Sections 1.2.1, 1.2.2, 1.2.3 (until page 15)
28.02.2020 Definition of a pre-measure. Examples. Carathéodory-Hahn extension. Proof of Theorems 1.2.20 and 1.2.21 Section 1.2.3 (until page 19)
4.03.2020 (recorded lecture) Repetition of the Proof of Theorem 1.2.21, n-dimensional intervals, elementary sets, volume of elementary sets, dyadic decomposition. Section 1.3 until page 21
6.03.2020 (recorded lecture) Proof of Lemma 1.3.4, Lebesgue measure and its regularity properties (Borel measure, approximation of a Lebesgue measurable set from inside and outside by closed and open sets). Comparison between Jordan measure and Lebesgue measure. An example of a set which is not Jordan measure (see for instance in T.Tao's book Remark 1.2.8) Section 1.3 until page 27 Class Notes 4-6.03.2020
11.03.2020 (recorded lecture by D-Camera) Proof of the fact that the set of rational number in [0,1] is not Jordan Measurable. The Lebesgue measure is Borel regular. Example of a set which is not Lebesgue measurable: Vitali Set. End of Section 1.3.and Section 1.4. For curiosity:an example of Lebesgue measurable set in R which is not Borel
13.03.2020 (recorded lecture by D-Camera) Vitali set continued. Exercise 1.4.1. Cantor Triadic set. The Lebesgue-Stieltjes measure on R and proof of some properties, metric measure, the Carathéodory criterium for a Borel measure Section 1.4, 1.5, 1.6, Class Notes 11-13.03.2020, Just for curiosity:Presentation of Cantor Set by Giada Franz
18.03.2020 (recorded lecture) Proof of Theorem 1.6.5. Introduction of Hausdorff measure For the proof of Theorem 1.6.5 see the proof of Satz 1.5.3 in Struwe's notes (it will be soon added in the Lecture Notes). Section 1.7 until page 38.
20.03.2020 (recorded lecture by D-Camera) Proof of Theorem 1.7.3 Proof of Lemma 1.7.5, Example 1.7.6 Definition of the Hausdorff dimension of a set of R^n, examples of sets with non-integer Hausdorff dimension Section 1.7. Class Notes 18-20.03.2020. Just for curiosity:Fractals and Prof. R. Mingione's talk about fractals (in italian),
25.03.2020 (recorded lectur with Tablet) Canton Dust and its Hausdorff dimension. Radon measures on R^n and their regularity properties Section 1.7 and 1.8 Class Notes 25.03.2020
27.03.2020 (recorded lecture with Tablet) Measurable functions: definition and properties. Approximation of nonnegative measurable functions by simple functions. Proof of Egoroff's Theorem and Lusin 's Theorem. Sections 2.1, 2.2, 2.3 until page 59. Class Notes 27.03.2020
1.04.2020 Convergence in measure and the relation with a.e. convergence. Definition of the integral with respect to a given Radon measure on R^n Section 2.4 and Section 3.1 until page 66, Class Notes 1.04.2020
3.04.2020 (recorded lecture with Tablet) Proof of Propositions 3.1.6, 3.1.8, 3.1.9, 3.1.10, Tchebicev Inequality and its consequences. Summable functions and some properties. Section 3.1 until page 70, Class Notes 3.04.2020
8.04.2020 (recorded lecture with Tablet) Proof of Theorem 3.1.15 about linearity of the integral. Proof of Corollary 3.1.16 and of Lemma 3.1.17 Section 3.1. until page 74. For the proof of Lemma 3.1.17 see Class Notes and the proof of Lemma 3.1.1 in Struwe's notes, Class Notes 8.04.2020
Frohe Ostern
22.04.2020 (recorded lecture with Tablet) Proof of Corollary 3.1.18 and of Proposition 3.1.19. Comparison betweem Riemann and Lebesgue integral. Proof of Fatou's Lemma and of Beppo Levi's Theorem. Derivation of Fatou's Lemma from Beppo Levi's Theorem. Proof of Fatou's Lemma through Beppo Levi's Theorem Section 3.1, 3.2, 3.3 until page 80, Class Notes 22.04.2020
24.04.2020 (recorded lecture with Tablet) Proof Dominated Convergence Theorem. Alternative Proof of Corollary 3.1.14 (see Class Notes). Differentiation of integrals depending on a parameter. Absolute continuity of Integrals. Proof of Theorem 3.5.3, Proof of Vitali 's Theorem. Section 3.3,3.4,3.5, 3.6.Class Notes 24.04.2020
29.04.2020 (recorded lecture with Tablet) Some consequences of Vitali's Theorem: proof of Theorem 3.6.5 and Theorem 3.6.6. Relation between Lebesgue summabililty and absolutely Riemann integrability in the generalized sense Section 3.6,Class Notes 29.04.2020
06.05.2020 (recorded lecture with Tablet) L^p spaces, Young Inequality, Hölder Inequality, Generalized Hölder Inequality (Cor. 3.7.8) Minkoswski Inequality. Section 3.7 until page 97. Class Notes 06.05.2020
8.05.2020 (recorded lecture with Tablet) Completeness and separability of L^p spaces. Proofs of Lemma 3.7.13, Theorem 3.7.15, Theorem 3.7.21 Section 3.7 Class Notes 08.05.2020
13.05.2020 (recorded lecture with Tablet) A criterium of convergence in the L^p spaces: Proof of Theorem 3.7.21. Tonelli's Theorem for series. Two counter-examples Section 3.7 .Theorem 0.0.2 in T. Tao's book.Class Notes 13.05.2020
15.05.2020 (recorded lecture with Tablet) Statement of Fubini and Tonelli's Theorems (no proof). Some applications. Convolution. Proof of Theorem 4.3.3 Section 4.1, 4.2 and 4.3 until page 122.Class Notes 15.05.2020
20.05.2020 (recorded lecture with Tablet) Proof of Corollary 4.3.4 and Proposition 4.3.6 (regularation property of the convolution) Section 4.3 until page 125.Class Notes 20.05.2020
22.05.2020 (recorded lecture with Tablet) Application of convolution: solution to the Laplace equation. Representation of a smooth function with compact support in terms of its Laplacian. Proof of Theorem 4.4.2 Section 4.4. For the proof of Theorem 4.4.2 see either Theorem 4.2.3 in Struwe'Notes or Theorem 3.2.1 in these lecture notes Class Notes 22.05.2020
27.05.2020 (recorded lecture with Tablet) Differentiation of Lebesgue measure. Proof of Theorem 5.1.2 and of Corollary 5.1.7 Hardy Maximal Function (definition and some properties and counter-examples) Section 5.1 until page 132. Class Notes 27.05.2020
29.05.2020 (recorded lecture with Tablet) Application of Theorem 5.1.2 (see class notes). Proof of Theorem 5.1.9 (Vitali's covering theorem for Lebesgue measure). Proof o f Proposition 5.1.6 (weak-L^1 property of the Hardy-Littlewood maximal function Section 5.1. Class Notes 29.05.2020
** Program and examples of questions for the exam *****

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