Program Overview

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Analysis III (Measure Theory)

Overview

Analysis III (Measure Theory) is a university program offered during the Autumn Semester 2025 by the D-MATH department. The program is led by Lecturer Francesca Da Lio and Coordinator Antonio Marini.

Lectures

• The lectures take place on Wednesdays from 15-16 in HG E5 and Fridays from 10-12 in HG G3.
• The first lecture is scheduled for Wednesday, September 17.

Course Description

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

Learning Outcomes

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 Rn 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 Lp spaces and their properties, including inequalities and completeness.
9. Write clear, rigorous proofs and explanations involving measurable sets, functions, and integrals.

Exercise Classes

• The first exercise classes will take place on Monday, September 22.
• Students are expected to enroll in a group on MyStudies.
• The groups and their details are as follows:
  • G-01: Mo 16-18, ML H 41.1, Johannes Luber, German
  • G-02: Mo 16-18, ML F 40, Malte Voos, German
  • G-03: Mo 16-18, LEE D 101, Elia von Salis, German
  • G-04: Mo 16-18, LFW C 5, Lucile Chapuis, English
  • G-05: Mo 16-18, CHN G 42, Raphael Angst, English

Bonus Exercises

• Each week, a "bonus" exercise will be available on the course page.
• The total number of bonus exercises for this semester is 13, each worth 1 point.
• The final bonus will consist of 2 multiple choice questions and will be worth 2 points.
• A grade bonus will be awarded according to the formula: Bonus = MIN(0.0125 × P, 0.125), where P is the number of points earned.

Exercise Sheets

• New exercises will be posted each Thursday.
• Students are expected to look at the problems over the weekend and prepare questions for the exercise class on Monday.
• Exercises marked with a ★ sign are more challenging, and those marked with a ◊ sign are particularly recommended for exam preparation.

Office Hours

• Students are encouraged to provide direct feedback and ask questions to the lecturer, course coordinator, and teaching assistants.
• The assistants from Group 3 offer regular office hours to address questions related to the courses and exercise sessions coordinated by Group 3.

Preparation for the Exam

• The course unit Analysis III is examined together with Analysis IV.
• The written exam lasts 180 minutes and will take place in the exam session of summer 2025 (alternatively winter 2026).
• A list of exercises for exam preparation, along with past exam questions, is provided.

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.