# Mass und Integral Spring 2021

Lecturer
Francesca Da Lio
Coordinator
Lectures
Wed 9-10
Fri 10-12
Exercise classes
Wed 10-12
Course Catalogue
401-2284-00L Mass und Integral

## Summer office hours

We are organizing some office hours before the summer exams. Here is the updated schedule.

Date Time Assistant Room
Wed 21.07. 14-15 Giada Franz HG F 33.1
Wed 28.07. 14-15 Michael Vogel ETH Zoom - 643 7419 1093
Wed 04.08. 14-15 Riccardo Plati HG F 33.1

## Lectures

The diary of the lectures is available here. You can find the lecture notes in this polybox folder and the links to the lecture recordings in the table below. The password is the one we sent you by email.

Date Content Reference (in the Lecture Notes) Video recording
1 Wed 24.02. Presentation of the course, preliminary notations and definitions. Section 1.1.1 Video - L01
2 Fri 26.02. Algebras, $$\sigma$$-algebras, $$\sigma$$-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. Section 1.1.2 Video - L02
3 Wed 03.03. Proof of Theorem 1.2.10 (the set of measurable sets is a $$\sigma$$-algebra), definition of a measure Space, Exercise 1.2.12. Section 1.2.2 Video - L03
4 Fri 05.03. Proof of Theorem 1.2.13. Definition of a covering. Construction of a measure. Proof of Theorem 1.2.17. Definition of a pre-measure. Examples. Carathéodory-Hahn extension. Sections 1.2.1, 1.2.2, 1.2.3 (until page 17) Video - L04
5 Wed 10.03. Conclusion of the Proof of Theorems 1.2.20 (part iii) and proof of Theorem 1.2.21 about the uniqueness of the Carathéodory-Hahn extension. Remark 1.2.22. Section 1.2.3 (until page 19) Video - L05
6 Fri 12.03. Definition of intervals and elementary sets in $$\mathbb R^n$$. Definition of the volume of an elementary set. The volume is $$\sigma$$-additive. Dyadic decomposition in $$\mathbb R^n$$. Definition of Lebesgue measure. Proof of Lemma 1.3.4 and Theorem 1.3.5. Proof of the fact that the Lebesgue measure is a Borel measure. Section 1.3 (until page 23), Bemerkung 1.3.2 in Struwe's notes Video - L06
7 Wed 17.03. Approximation of a Lebesgue measurable set from inside and outside by closed and open sets. Proof of Theorem 1.3.8. Corollary 1.3.9 and Corollary 1.3.10 (this last one has been left as an exercise). Jordan Measure. Examples of sets which are Jordan measurables. Sections 1.3 and 1.4 (until page 28) Video - L07
8 Fri 19.03. Proof of Theorem 1.4.1. Examples of sets which are not Jordan measurable. The Lebesgue measure is Borel regular. Example of a set which is not Lebesgue measurable: Vitali Set. Sections 1.4 and 1.5 (until page 31) Video - L08
9 Wed 24.03. Every countable set in $$\mathbb R$$ has measure zero. Description of the Cantor triadic set: example of an uncountable with zero Lebesgue measure. Its representation in base $$b=3$$. Sections 1.5 and 1.6 Video - L09
10 Fri 26.03. The Lebesgue-Stieltjes measure on $$\mathbb R$$ and proof of some regularity properties, definition of a metric measure, the Carathéodory criterium for a Borel measure: proof of Theorems 1.7.2, 17.4, 1.7.5. Section 1.7 Video - L10
11 Wed 31.03. Introduction of Hausdorff measure. Proof of the fact that the zero dimensional Hausdorff measure is the counting measure. Proof of Theorem 1.8.3 (the s-dimensional Hausdorff measure is a Borel regular measure). Section 1.8 (until page 41) Video - L11
12 Wed 14.04. Proof of Theorem 1.8.3 (continued), proof of Lemma 1.8.5, Example 1.8.6, definition of the Hausdorff dimension of a set of $$\mathbb R^n$$. Section 1.8 (until page 48) Video - L12
13 Fri 16.04. Some remarks about the Hausdorff dimension of subsets of $$\mathbb R^n$$, description of Cantor dust, Radon measures, proof of Theorem 1.9.3, definition of measurable functions. Sections 1.8.1, 1.9 and 2.1 (until page 56) Video - L13
14 Wed 21.04. Measurable functions: equivalent definitions (see also Exercise 6.2). Proof of Theorem 2.2.5. Composition of a measureble function $$f:A\to \mathbb R$$ with a Borel function $$g:\mathbb R \to \mathbb R$$ is measurable. Comparison with Exercise 6.5. Sections 2.2 (until page 58) Video - L14
15 Fri 23.04. Approximation of nonnegative measurable functions by simple functions. Proof of Egoroff's Theorem and Lusin 's Theorem. Sections 2.2 and 2.3 (until page 65) Video - L15
16 Wed 28.04. Proof of Lusin 's Theorem (continued), Convergegence in measure. Relation between convergence in measure and the almost everywhere convergence. Sections 2.3 and 2.4 (until page 66) Video - L16
17 Fri 30.04. Definition of the integral with respect to a given Radon measure on $$\mathbb R^n$$. Proof of Propositions 3.1.7, 3.1.9, 3.1.10, 3.1.11, Tchebicev Inequality and its consequences. Summable functions and some properties. Section 3.1 (until page 75) Video - L17
18 Wed 05.05. Proof of Theorem 3.1.15 about linearity of the integral. Proof of Corollaries 3.1.16 , 3.1.18 and of Proposition 3.1.19. Section 3.1 (until page 79) Video - L18
19 Fri 07.05. Proof of Lemma 3.1.17, comparison betweem Riemann and Lebesgue integral. Example of a function which is not Riemann integral. Proof of Fatou's Lemma and of Beppo Levi's Theorem. Application: integral of a series of measurable nonnegative functions is the sum of of the integrals. Sections 3.2 and 3.3 (until page 87) Video - L19
20 Wed 12.05. Proof of Dominated Convergence Theorem. Two different proofs of Corollary 3.1.14. Absolute continuity of integrals. Sections 3.3 and 3.5 (for the proof of Corollary 3.1.14 see also the class notes). Video - L20
21 Fri 14.05. Proof of Vitali Theorem and of Theorems 3.6.5 and 3.6.6. Introduction of $$L^p$$ spaces. Examples. Sections 3.6 and 3.7 (until page 100) Video - L21
22 Wed 19.05. Proof of Young inequality, Hölder inequality, Minkowski inequality. Section 3.7 (until page 105) Video - L22
23 Fri 21.05. Proof of the completeness of $$L^p$$ spaces, for $$p \in [1,\infty]$$. Tonelli theorem for the series. Product measures and Fubini and Tonelli theorems (only statements). Examples and Remarks. Section 3.7 (until page 108) and Section 4.1 (until page 126) Video - L23
24 Wed 26.05. Applications to Fubini and Tonelli Theorems. Definition of the convolution. Sections 4.2 and 4.3 (until page 129) Video - L24
25 Fri 28.05. Proof of Theorem 4.3.3, Corollary 4.3.4, Theorem 4.3.6. Section 4.3 Video - L25
26 Wed 02.06. Separability of $$L^p$$ spaces. Proofs of the first part of Theorem 3.7.15. Remarks on $$L^{\infty}$$ space. Section 3.7 (until page 110) Video - L26
27 Fri 04.06. Second part of proof of Theorem 3.7.15: density of the space of continuous functions with compact support in the $$L^p$$ spaces for $$1\le p<{\infty}$$. Proof of Theorem 3.7.21. Section 3.7 Video - L27

## Forum

In order to easily interact, we set up a forum for our course at the link Mass und Integral (Spring 2021) - Forum. You have to sign up with your ETH credentials. There you find several topics where you can ask questions and discuss about the lectures, the problem sets, the exam, etc.

## Exercise classes

Please register and enroll for a teaching assistant in myStudies. The enrollment is needed to attend the exercise class and to hand in your homework.

ZeitRoomTutorLanguage
Wed 10-12ETH Zoom - 940 7940 4722Maran MohanaranganEnglish
Wed 10-12ETH Zoom - 962 7393 5368Riccardo PlatiEnglish
Wed 10-12ETH Zoom - 986 5790 1652Salome SchumacherGerman
Wed 10-12ETH Zoom - 920 6767 2168Leon StaresinicEnglish
Wed 10-12ETH Zoom - 977 0647 9688Michael VogelGerman

The exercise classes are not recorded, thus we suggest you to attend the live exercise classes on zoom. You can find the notes of Salome's exercise classes in this polybox folder, the password is the same as the one for the lecture notes. Moreover in the table below you can find the notes of Maran's exercise classes.

Date Content Notes
1 Wed 03.03. The motivation behind $$\sigma$$-algebras Notes - E01
2 Wed 10.03. Abstract measure spaces Notes - E02
3 Wed 17.03. Probability measures and the Lebesgue measure Notes - E03
4 Wed 24.03. A Lebesgue measurable set that is not Borel Notes - E04
5 Wed 31.03. Fractals and the Hausdorff dimension Notes - E05
6 Wed 14.04. Measurable functions Notes - E06
7 Wed 21.04. Almost everywhere Notes - E07
8 Wed 29.04. Littlewood’s three principles Notes - E08
9 Wed 05.05. Lebesgue integration theory Notes - E09
10 Wed 12.05. Convergence results and absolute continuity Notes - E10
11 Wed 19.05. Modes of convergence Notes - E11
12 Wed 26.05. $$L^p$$ spaces Notes - E12
13 Wed 02.06. The fundamental theorem of calculus for Lebesgue integrals Notes - E13

## Exercises

The submission of your solutions to the exercises should be done via the platform SAMUpTool, where you can access also your corrected exercise sheets. Please register in myStudies to be able to access the submission platform.

A new exercise sheet is uploaded here every Monday and you have time until Wednesday at 10am of the following week to hand in your solutions.

Assignment date Exercise sheet Due date Solutions
Mon 01.03. Serie 1 (German version) Wed 10.03. Solutions 1 (German version)
Mon 08.03. Serie 2 (German version) Wed 17.03. Solutions 2 (German version)
Mon 15.03. Serie 3 (German version) Wed 24.03. Solutions 3 (German version)
Mon 22.03. Serie 4 (German version) Wed 31.03. Solutions 4 (German version)
Mon 29.03. Serie 5 (German version) Wed 14.04. Solutions 5 (German version)
Mon 12.04. Serie 6 (German version) Wed 21.04. Solutions 6 (German version)
Mon 19.04. Serie 7 (German version) Wed 28.04. Solutions 7 (German version)
Mon 26.04. Serie 8 (German version) Wed 05.05. Solutions 8 (German version)
Mon 03.05. Serie 9 (German version) Wed 12.05. Solutions 9 (German version)
Mon 10.05. Serie 10 (German version) Wed 19.05. Solutions 10 (German version)
Mon 17.05. Serie 11 (German version) Wed 26.05. Solutions 11 (German version)
Mon 24.05. Serie 12 (German version) Wed 02.06. Solutions 12 (German version)
Mon 31.05. Serie 13 (German version) Wed 09.06. Solutions 13 (German version)

## Further References

• Lawrence Evans and Ronald Gariepy, Measure Theory and Fine Properties of Functions, Textbooks in Mathematics, CRC Press, 2015.
• Walter Rudin, Real and Complex Analysis, Higher Mathematics Series, 3. Edition, McGraw-Hill, 1986.
• Robert Bartle, The Elements of Integration and Lebesgue Measure, Wiley Classics Library, John Wiley & Sons, 1995.
• Michael Struwe, Analysis III: Mass und Integral, Lecture Notes, ETH Zürich, 2013.
• Urs Lang, Mass und Integral, Lecture Notes, ETH Zürich, 2018.
• P. Cannarsa and T. 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