This course is a continuation of "Real Analysis I" in the first quarter. In this course, we deal with more advanced concepts and properties of measures and integration by means of measures (Lebesgue integration). We first explain construction and extension of measure. Second, we show the relation between Lebesgue integral and Riemann integral. Third, we introduce function spaces defined by means of integration and studies their basic properties. Finally, we study the Fubini theorem as a measure-theoretic treatment of (iterated) integral on product spaces.
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 able to explain the outline of basic construction of measures
Be able to explain the difference between Lebesgue integration and Riemann integration.
Be able to apply the theory of Lebesgue integration to problems in calculus.
Be familiar with the notion of functional inequalities in integration and function spaces defined by integration.
Be able to apply the Fubini theorem to calculate multiple integrals and iterated integrals correctly.
Hopf's extension theorem, outer measure, Caratheodory measurability, Dynkin system therem, Riemann integral, H\"older's inequality, Minkowski's inequality, Lebesgue space, product measure, Fubini theorem
Intercultural skills | Communication skills | ✔ Specialist skills | Critical thinking skills | Practical and/or problem-solving skills |
Alternation of standard lecture course and problem session.
Course schedule | Required learning | |
---|---|---|
Class 1 | Extension theorem for measures | Details will be provided during each class session |
Class 2 | Problem session | Details will be provided during each class session |
Class 3 | Outer measures and construction of measures | Details will be provided during each class session |
Class 4 | Problem session | Details will be provided during each class session |
Class 5 | Relation between Riemann integral and Lebesgue integral | Details will be provided during each class session |
Class 6 | Problem session | Details will be provided during each class session |
Class 7 | L^p-spaces and its completeness, fundamental functional inequalities | Details will be provided during each class session |
Class 8 | Problem session | Details will be provided during each class session |
Class 9 | Product measure and iterated integral | Details will be provided during each class session |
Class 10 | Problem session | Details will be provided during each class session |
Class 11 | Fubini theorem and its applications | Details will be provided during each class session |
Class 12 | Problem session | Details will be provided during each class session |
Class 13 | Extension of Fubini theorem | Details will be provided during each class session |
Class 14 | Problem session | Details will be provided during each class session |
None required.
W. Rudin "Real and complex analysis" McGraw-Hill.
H. Royden "Real Analysis"
Final exam (about 50%), problem session (about 50%).
Student are required to have passed Real Analysis I.
Students are expected to have passed Introduction to Analysis I+II and Introduction to Topology I+II.