In this course, we deal with 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. We next study convergence theorems. They would be a basis of integration theory as well as application of Lebesgue integration. We next explain construction and extension of measure. We next show the relation between Lebesgue integral and Riemann integral. Then, 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. We strongly recommend to take this course with "Exercises in Analysis C I".
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.
Get to 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.
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.
sigma-algebra, measurable space, measure, measure space, Lebesgue measure, measurable function, Lebesgue integration, monotone convergence theorem, Fatou's lemma, dominated convergence theorem, Hopf's extension theorem, outer measure, Caratheodory measurability, Dynkin system theorem, 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 |
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- | - | ✔ | - | - |
Standard lecture course.
Course schedule | Required learning | |
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Class 1 | Overview of measure theory and Lebesgue integration | Details will be provided during each class session |
Class 2 | Sigma-algebra | Details will be provided during each class session |
Class 3 | (Countably additive) measure and its basic properties, completeness | Details will be provided during each class session |
Class 4 | Measurable functions | Details will be provided during each class session |
Class 5 | Definition of integral and its basic properties | Details will be provided during each class session |
Class 6 | Convergence theorems (Monotone convergence theorem, Fatou's lemma and dominated convergence theorem ) and examples | Details will be provided during each class session |
Class 7 | Applications of convergence theorems | Details will be provided during each class session |
Class 8 | Extension theorem for measures | Details will be provided during each class session |
Class 9 | Outer measures and construction of measures | Details will be provided during each class session |
Class 10 | Relation between Riemann integral and Lebesgue integral | Details will be provided during each class session |
Class 11 | L^p-spaces and its completeness, fundamental functional inequalities | Details will be provided during each class session |
Class 12 | Product measure and iterated integral | Details will be provided during each class session |
Class 13 | Fubini theorem and its applications | Details will be provided during each class session |
Class 14 | Extension of Fubini theorem | 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.
Midterm exam (about 50%), final exam (about 50%).
Students are expected to have passed Introduction to Analysis I+II and Introduction to Topology I+II.
Strongly recommended to take ZUA.C306: Exercises in Analysis C I (if not passed yet) at the same time.