We give an introduction on measures and Lebesgue integration, which are notions refining and extending area and integration. After introducing Lebesgue (outer) measure and measurable functions in the 1-dimensional Euclidean space, we define the Lebesgue integral and explain the fundamental properties.
We present convergence theorems concerning exchange between limit and integration for sequences of functions.
Next we extend the theory of Lebesgue integration to multi dimensional Euclidean spaces and present Fubini's theorem
on the relation between the multiple integral and the iterated integral.
Finally, we deal with the Lebesgue spaces and related inequalities for functions.
Understand the fundamentals on the theory of measures and Lebesgue integration. Apply convergence theorems and Fubini's theorem to specific problems.
Outer measure, measurable sets, measure, measurable functions, Lebesgue integral, convergence theorems, Fubini's theorem
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
For understanding of this course, it is necessary to be skilled at the contents by the calculation by hand. Therefore the exercise class is given every two weeks.
Course schedule | Required learning | |
---|---|---|
Class 1 | Overview on the theory of Lebesgue integral | Understand the contents of the lecture. |
Class 2 | Lebesgue outer measure | Understand the contents of the lecture. |
Class 3 | Exercise for preliminaries and Lebesgue outer measure | Cultivate a better understanding of lectures. |
Class 4 | Lebesgue measurable set | Understand the contents of the lecture. |
Class 5 | Definition on Lebesgue measurable functions | Understand the contents of the lecture. |
Class 6 | Exercise for Lebesgue measurable set and Lebesgue measurable functions | Cultivate a better understanding of lectures. |
Class 7 | Properties on Lebesgue measurable functions | Understand the contents of the lecture. |
Class 8 | Definition of Lebesgue integral | Understand the contents of the lecture. |
Class 9 | Exercise for Lebesgue measurable functions and Lebesgue integral | Cultivate a better understanding of lectures. |
Class 10 | Properties of Lebesgue integral | Understand the contents of the lecture. |
Class 11 | Convergence theorems | Understand the contents of the lecture. |
Class 12 | Exercise for Lebesgue integral and convergence theorems | Cultivate a better understanding of lectures. |
Class 13 | Convergence theorems | Understand the contents of the lecture. |
Class 14 | Relation between Lebesgue integral and Riemann integral | Understand the contents of the lecture. |
Class 15 | Exercise for convergence theorems and relation between Lebesgue integral and Riemann integral | Cultivate a better understanding of lectures. |
Class 16 | Product measure | Understand the contents of the lecture. |
Class 17 | Fubini's theorem | Understand the contents of the lecture. |
Class 18 | Exercise for product measure and Fubini's theorem | Cultivate a better understanding of lectures. |
Class 19 | フビニの定理の応用 | Understand the contents of the lecture. |
Class 20 | Lebesgue spaces | Understand the contents of the lecture. |
Class 21 | Exercise for Fubini’s thoerem and Lebesgue spaces | Cultivate a better understanding of lectures. |
To enhance effective learning, students are encouraged to spend a certain length of time outside of class on preparation and review (including for assignments), as specified by the Tokyo Institute of Technology Rules on Undergraduate Learning (東京工業大学学修規程) and the Tokyo Institute of Technology Rules on Graduate Learning (東京工業大学大学院学修規程), for each class.
They should do so by referring to textbooks and other course material.
To be announced
To be announced.
By scores of the examination and the reports.
The students are encouraged to understand the limit, the supremum (infimum), and the fundamentals on sets and topology.