# Mass und Integral, D-MATH Spring Semester 2021

## Lecturer: Prof. Francesca Da Lio

## Exercise hours coordinator: Giada Franz

- Lecture Notes (These notes will be continuously upadated during the course)
**Class Notes****Program and examples of questions for the oral exam**- Course Webpage
- Further Information

## Diary of the lectures

Date | Content | Notes | Reference | |
---|---|---|---|---|

1 | 24.02.2021 | Slides of presentation of the course, Preliminary notations and definitions | Class Notes |
Section 1.1.1 in the Lecture Notes |

2 | 26.02.2021 | 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. | Class Notes |
Section 1.1.2 in the Lecture Notes |

3 | 03.03.2021 | Proof of Theorem 1.2.10 (the set of measurable sets is a sigma algebra), definition of a measure Space, Exercise 1.2.12. | Class Notes |
Section 1.2.2 in the Lecture Notes |

4 | 05.03.2021 | 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. Proof of Theorem 1.2,20 (part i) and ii)) | Class Notes |
Sections 1.2.1, 1.2.2, 1.2.3 (until page 17). |

5 | 10.03.2021 | 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 | Class Notes |
Section 1.2.3 (until page 19) |

6 | 12.03.2021 | Definition of intervals and elementary sets in R^n. Definition of the volume of an elementary set. The volume is sigma-additive. Dyadic decomposition in 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. | Class Notes |
Section 1.3 (until page 23). For the proof of the sigma additivity of the volume see "Bemerkung 1.3.2" in Struwe's notes and the class notes. |

7 | 17.03.2021 | 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. | Class Notes |
Section 1.3 & Section 1.4 until page 28 |

8 | 19.03.2021 | 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. | Class Notes |
Section 1.4 & Section 1.5 until page 31. For curiosity: 1) An example of Lebesgue measurable set in R which is not Borel, 2) The Banach-Tarski theorem 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 | 24.03.2021 | 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. |

10 | 26.03.2021 | The Lebesgue-Stieltjes measure on 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) | Class Notes |
Section 1.7 |

11 | 31.03.2021 | 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) | Class Notes |
Section 1.8 until page 41. 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 |

** | ** | Frohe Ostern |
** | ** |

12 | 14.04.2021 | 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 R^n. | Class Notes |
Section 1.8 until page 48 |

13 | 16.04.2021 | Some remarks about the Hausdorff dimension of subsets of R^n, description of Cantor dust, Radon Measures, Proof of Theorem 1.9.3, definition of measurable functions | Class Notes |
Section 1.8.1, 1.9 and 2.1 until page 56 |

14 | 21.04.2021 | Measurable functions: equivalent definitions (see also Serie 6 ex 2). Proof of Theorem 2.2.5. Composition of a measureble function f:A->IR with a Borel function g:IR -> IR is measurable. Comparison with exercise 5 in Serie 6. | Class Notes |
Section 2.2 until page 58. |

15 | 23.04.2021 | Approximation of nonnegative measurable functions by simple functions. Proof of Egoroff's Theorem and Lusin 's Theorem. | Class Notes |
Section 2.2 & 2.3 until page 65. Proof of Remark 2.2.8 (by Jeremy Feusi) |

16 | 28.04.2021 | Proof of Lusin 's Theorem (continued), Convergegence in measure. Relation between convergence in measure and the almost everywhere convergence. | Class Notes |
Section 2.3 & 2.4 until page 66 |

17 | 30.04.2021 | Definition of the integral with respect to a given Radon measure on 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. | Class Notes |
Section 3.1 until page 75 |

18 | 5.05.2021 | 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. | Class Notes |
Section 3.1 until page 79 |

19 | 7.05.2021 | Proof of Lemma 3.1.17, Comparison betweem Riemann and Lebesgue integrals. Example of a function which is not Riemann integrable. Proof of Fatou's Lemma and of Beppo Levi's Theorem. Application: integral of a series of measurable nonnegative functions is the sum of the integrals. | Class Notes |
Section 3.2 and 3.3 until page 87. |

20 | 12.05.2021 | Proof of Dominated Convergence Theorem. Two different proofs of Corollary 3.1.14. Absolute continuity of integrals. | Class Notes |
End of Section 3.3 and Section 3.5. For the proof of Corollary 3.1.14 see also the class notes |

21 | 14.05.2021 | Proof of Vitali Theorem and of Theorem 3.6.5 & Theorem 3.6.6. Introduction of L^p spaces. Examples. | Class Notes |
Sections 3.6 and 3.7 until page 100. |

22 | 19.05.2021 | Proof of Young Inequality, Hölder Inequality, Minkowski Inequality | Class Notes |
Section 3.7 until page 105 |

23 | 21.05.2021 | Proof of the completeness of L^p spaces, for p in [1,\infty]. Tonelli theorem for the series. Product measures and Fubini and Tonelli Theorem (only statements). Examples and Remarks. | Class Notes |
Section 3.7 until page 108 and Section 4.1 until page 126 |

24 | 26.05.2021 | Applications to Fubini and Tonelli Theorems. Definition of the convolution. | Class Notes |
Sections 4.2 and 4.3 until page 129. |

25 | 28.05.2021 | Proof of Theorem 4.3.3, Corollary 4.3.4, Theorem 4.3.6 | Class Notes |
Sections 4.3 |

26 | 2.06.2021 | Separability of L^p spaces. Proofs of the first part of Theorem 3.7.15. Remarks on L^{\infty} space. | Class Notes |
Sections 3.7 until page 110. For the details of the proof of the fact that the L^{\infty} space is not in general separable see also the class notes. |

27 | 4.06.2021 | 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. | Class Notes |
Sections 3.7 until the end. |

** |
Viel Erfolg!! |
***** |

## Recommended bibliography (Undergraduate-Master level):

**Lawrence Evans and Ronald Gariepy**, Measure Theory and Fine Properties of Functions, Textbooks in Mathematics, CRC Press, 2015.**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.**Piermarco Cannarsa and Teresa 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.**Further reading:** **W.F. Eberlein**,*Notes on Integration I: The Underlying Convergence Theorem*, Comm. Pure Appl. Math. 10 (1957), 357–360.- Ask yourself dumb questions – and answer them! (by Terence Tao);
- How to write Mathematics (by Paul R. Halmos)