In this class, I will explain the Lebesgue integral. The Lebesgue integral is a standard integral in modern analysis and is
necessary for studies in functional analysis, differential equations, probability, quantum mechanics and so on. In the introduction
of the class, I will indicate weak points of the Riemann integral and describe necessity to introduce a new integral (the
Lebesgue integral). Next, after the fashion of F.Riesz, we will define the Lebesgue integral as the limit of integrals for
the sequences of step functions. Then, we will show that the new integral is an extension of the Riemann integral and satisfies
theorems of term-wise integral under weaker conditions than that of the Riemann integral. Also, making use of the theorems,
we define the "measure" of a set, which is a generalization of the length, area and volume of the set. In fact, Lebesgue himself
defined the measure first and constructed his integral on the basis of the measure. In the final class, I will describe that
the integrals and the measures defined after the fashion of Riesz and Lebesgue, respectively, completely correspond with each
other.
The purpose of the class is to master the Lebesgue integral, particularly, the theorems of term-wise integration, which is
necessary for learning functional analysis, differential equations, probability and so on.
- To recognize differences between the Lebesgue integral and the Riemann integral.
- To be able to use the theorems of term-wise integration.
- To understand the measurabilities of sets and of functions.
| Class schedule | HW assignments (Including preparation and review of the class.) | Amount of Time Required | |
|---|---|---|---|
| 1. | Why is the Lebesgue integral necessary ? | Check out "(Riemann) integral" in "Mathematics I". | 190minutes |
| 2. | Lebesgue integral (1) Integration of step function, Sets with measure zero | Review the previous session in advance. | 190minutes |
| 3. | Lebesgue integral (2) Extension of integral of step function, Properties of L^+ function | Review the previous session in advance. | 190minutes |
| 4. | Lebesgue integral (3) Definition, Relation with Riemann integral | Review the previous session in advance. | 190minutes |
| 5. | Theorem of term-wise integration (1) Beppo Levi's theorem | Review the previous session in advance. | 190minutes |
| 6. | Theorem of term-wise integration (2) Lebesgue's convergence theorem | Review the previous session in advance. | 190minutes |
| 7. | Theorem of term-wise integration (3) Fatou's lemma, Relation with improper Riemann integral | Review the previous session in advance. | 190minutes |
| 8. | Mid-term exam | Review Sessions 1-7 in advance. | 190minutes |
| 9. | Integral of multivariable functions and Fubini's theorem | Review Sessions 1-7 in advance. | 190minutes |
| 10. | Measure theory (1) Measurable functions | Check out "integrability" in Session 5 in advance. | 190minutes |
| 11. | Measure theory (2) Measurable sets, Relation between measurable functions and measurable sets | Review the previous session in advance. | 190minutes |
| 12. | Measure theory (3) Measurabilities of open sets and Borel sets | Review the previous session in advance. | 190minutes |
| 13. | Definition of integration by Lebesgue (from measure to integral) | Review the previous session in advance. | 190minutes |
| 14. | Final exam | Review Sessions 9-13 in advance. | 190minutes |
| Total. | - | - | 2660minutes |
| Mid-term exam | Final exam | Total. | |
|---|---|---|---|
| 1. | 25% | 25% | 50% |
| 2. | 25% | 5% | 30% |
| 3. | 20% | 20% | |
| Total. | 50% | 50% | - |
Read Chapters 1 and 2. In particular, check out "supremum", "infimum", "limit", "superior limit", "inferior limit", "continuity",
"uniform continuity" and "uniform convergence". Check out "(Riemann) integral" in "Mathematics I".
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Last modified : Wed Oct 17 07:33:16 JST 2018