Measures and integration
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. Topics include:
- Abstract measure spaces
- Lebesgue integration
- Convergence theorems
- Product measures and Fubini's theorem
- L^p spaces
- Borel measures on locally compact Hausdorff spaces
- Decomposition and differentiation of measures
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 homework problems outside of class meetings
Assessment methods
Written homeworks and a written exam
Supervision
| Office hours | No |
| 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.
- E. M. Stein and R. Shakarchi, Real Analysis, Princeton University Press, Princeton, NJ, 2005.
- T. Tao, Introduction to Measure Theory, American Mathematical Society, Providence, RI, 2011.
Notes/Handbook
Lecture notes will be provided
Moodle Link
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Measures and integration
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Measures and integration
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Measures and integration
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional