In this course, we deal with basic concepts and properties of measures and integration by means of measures (Lebesgue integral). We first explain basics of sigma-algebra and (countably additive) measure. It includes the study of Lebesgue measures which are one of the most fundamental measures. We next introduce measurable functions, which are candidates of integrand, and Lebesgue integration, and study their elementary properties. Finally, we study convergence theorems.
They would be a basis of integration theory as well as application of Lebesgue integration. This course will be succeeded by "Real Analysis II" in the second quarter.
The theory of measures and integrations was constructed by Lebesgue on the basis of set theory. These concepts are a natural extension of length, area, volume and probability etc. We can naturally handle operations involving infinity (e.g. limit for figures and functions) within the framework of this theory. In this course, we would like to address how the notion of integration is extended by Lebesgue integration and how effective it is in analysis.
Students are expected to:
Be familiar with the notion of sigma-algebra and measure.
Be able to explain the reason why given measurable functions are measurable.
Know the reason why elementary property of integration holds and be able to use them freely.
Be able to apply convergence theorems by checking their assumptions correctly.
sigma-algebra, measurable space, measure, measure space, Lebesgue measure, measurable function, Lebesgue integration, monotone convergence theorem, Fatou's lemma, dominated convergence theorem
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Alternation of standard lecture course and problem session.
Course schedule | Required learning | |
---|---|---|
Class 1 | Overview of measure theory and Lebesgue integration | Details will be provided during each class session |
Class 2 | Problem session | Details will be provided during each class session |
Class 3 | Sigma-algebra | Details will be provided during each class session |
Class 4 | Problem session | Details will be provided during each class session |
Class 5 | (Countably additive) measure and its basic properties, completeness | Details will be provided during each class session |
Class 6 | Problem session | Details will be provided during each class session |
Class 7 | Measurable functions | Details will be provided during each class session |
Class 8 | Problem session | Details will be provided during each class session |
Class 9 | Definition of integral and its basic properties | Details will be provided during each class session |
Class 10 | Problem session | Details will be provided during each class session |
Class 11 | Convergence theorems (Monotone convergence theorem, Fatou's lemma and dominated convergence theorem ) and examples | Details will be provided during each class session |
Class 12 | Problem session | Details will be provided during each class session |
Class 13 | Applications of convergence theorems | Details will be provided during each class session |
Class 14 | Problem session | Details will be provided during each class session |
Class 15 | Evaluation of progress | Details will be provided during each class session |
None required.
W. Rudin "Real and complex analysis" McGraw-Hill.
Final exam (about 50%), problem session (about 50%).
Students are expected to have passed Introduction to Analysis I+II and Introduction to Topology I+II.