Measure theory
Summary
This course provides an introduction to the theory of measures and integration on abstract measure spaces.
Content
Measure theory serves as a foundation for many areas of modern analysis, such as harmonic analysis, functional analysis, probability theory, and ergodic theory. This course focuses on buliding up the general framework and introducing frequently-used tools in measure theory, accompanied by motivating problems and examples from other areas of mathematics. Topics include:
- Measures on abstract spaces
- Integration of measurable functions
- Convergence theorems for integrals (Fatou's lemma, monotone convergence, dominated convergence)
- Measures on the real line (Lebesgue-Stieltjes measures)
- Borel measures on locally compact Hausdorff spaces (Riesz representation theorem)
- Product measures and Fubini's theorem
- L^p spaces
- Decomposition and differentiation of measures, density functions
Keywords
analysis, measure theory, Lebesgue integration, L^p spaces
Learning Outcomes
By the end of the course, the student must be able to:
- Define fundamental objects such as sigma-algebras, measures, measurable functions, etc.
- Apply the main theorems to problems in analysis and other areas
- Prove results in measure theory
- Identify common proof techniques used in analysis
Transversal skills
- Use a work methodology appropriate to the task.
- Continue to work through difficulties or initial failure to find optimal solutions.
- Demonstrate a capacity for creativity.
- Demonstrate the capacity for critical thinking
Teaching methods
Weekly lectures and exercise sessions
Expected student activities
Participate in lectures and exercise sessions and complete assigned problems outside of class meetings
Assessment methods
Written homeworks and small quizzes during the semester, written final exam
Supervision
Assistants | Yes |
Forum | Yes |
Resources
Bibliography
- G. B. Folland, Real Analysis (second edition), John Wiley & Sons, Inc., New York, 1999.
- W. Rudin, Real and Complex Analysis (third edition), McGraw-Hill Book Co., New York, 1987
- T. Tao, Introduction to Measure Theory, American Mathematical Society, Providence, RI, 2011.
Ressources en bibliothèque
Notes/Handbook
Lecture notes will be provided
Moodle Link
Dans les plans d'études
- Semestre: Automne
- Forme de l'examen: Ecrit (session d'hiver)
- Matière examinée: Measure theory
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Automne
- Forme de l'examen: Ecrit (session d'hiver)
- Matière examinée: Measure theory
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Automne
- Forme de l'examen: Ecrit (session d'hiver)
- Matière examinée: Measure theory
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel
- Semestre: Automne
- Forme de l'examen: Ecrit (session d'hiver)
- Matière examinée: Measure theory
- Cours: 2 Heure(s) hebdo x 14 semaines
- Exercices: 2 Heure(s) hebdo x 14 semaines
- Type: optionnel