SF2709 Integration Theory, 7.5 hp
Fall term 2009
Last news
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Program of the oral examination
- Homework assignement to 3 Dec:
4.4: 1,4.
- Homework assignement to 26 Nov:
4.2: 3,4,9;
4.3: 3.
- Homework assignement to 19 Nov:
4.1: 1,2,3.
- Homework assignement to 12 Nov:
3.3: 7, 8b, 9. In exercise 3.3.8b) you may assume for simplicity that
the function $\phi$ is differentiable.
- Homework assignement to 5 Nov:
3.1: 1, 2, 4.
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Homework assignement to 29 Okt:
2.4: 8, 10;
2.5: 4.
- Homework assignement to 15 Okt:
2.3: 7
2.4: 1, 2, 3.
- Homework assignement to 8 Okt:
2.3: 4, 6.
- Homework assignement to 1 Okt:
1.4: 3, 4;
2.1: 1, 3, 4.
- Homework assignement to 24 Sep:
1.4: 1, 2.
- Homework assignement to 17 Sep:
1.3: 1a, 2, 3, 7.
Solution of exercise 1.3.7
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Homework assignement to 10 Sep:
1.1: 1,2,4,6 and 1.2: 1ab,2,4.
Course information
- Course leader: Serguei Shimorin, 08-790 6692,
shimorin@math.kth.se
- Schedule:
The course holds during the fall term 2009 once per week on Thursdays at 13.15 - 15.00 in the room 3733
at the Department of Mathematics. Attention: there is no lecture 22 Oktober.
The course is given in English.
- Prerequisites:
Analys grundkurs SF2700.
- Course description:
The course goal is to learn about the notions of measure and Lebesgue integral.
Emphasis will be on the Lebesgue measure and integral on the real line and Euclidian spaces.
- Topics
Abstract measures. Measurable functions. Lebesgue integration.
Lebesgue measure. Functions of bounded variation
Signed measures, Jordan decomposition, absolutely continuous and singular measures,
Radon-Nykodim theorem, Lebesgue decomposition.
Fubini's theorem.
Normed and metric spaces. Lp spaces. Hölder ad Minkowski inequalities.
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Course litterature
- Donald L. Cohn, Measure Theory. Birkhäuser; Boston, Basel, Berlin.
- Examination: The examination consists of two parts: homework
assignements during the course and an oral test at the end. The homework
assignements are given after each lecture for a week and they should be delivered
to the lecturer at the next lecture for grading. To pass the course it is sufficient
(and necessary!)
to have 50% accepted homework exercises. This gives the grade "E". More than 80%
accepted exercises gives the grade "D". To get higher grade ("C", "B" or "A")
one should pass also the
oral test and the final grade will depend on it and the number of accepted homework
exercises.
For graduate students it is sufficient to have 50% accepted homework exercises
to pass the course.
Lecture plan (preliminary)
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3 Sep |
Lecture 1 |
| Algebras and sigma-algebras of sets. Measures. |
Section: | 1.1, 1.2. |
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10 Sep |
Lecture 2 |
| Outer measures and measures.
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Section: | 1.3 |
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17 Sep |
Lecture 3 |
| Lebesgue measure. Completeness and regularity properties. |
Section: | 1.4, 1.5. |
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24 Sep |
Lecture 4 |
| Measurable functions. |
Section: | 2.1, 2.2. |
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1 Okt |
Lecture 5 |
| The Lebesgue integral. |
Section: | 2.3, 2.6 |
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8 Okt |
Lecture 6 |
| Limit theorems. The Riemann integral. |
Section: | 2.4, 2.5. |
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15 Okt |
Lecture 7 |
| Modes of convergence. Lp spaces. |
Section: | 3.1, 3.3. |
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29 Okt |
Lecture 8 |
| Properties of Lp spaces. Normed spaces and duality. |
Section: | 3.2, 3.4, 3.5. |
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5 Nov |
Lecture 9 |
| Signed and complex measures. Decompositions. |
Section: | 4.1 |
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12 Nov |
Lecture 10 |
| Absolute continuity and singularity. |
Section: | 4.2, 4.3. |
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19 Nov |
Lecture 11 |
| Functions of boiunded variation. Duality of Lp spaces. |
Section: | 4.4, 4.5. |
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26 Nov |
Lecture 12 |
| Product measures. |
Section: | 5.1. |
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3 Dec |
Lecture 13 |
| Fubini's theorem. Applications. |
Section: | 5.2, 5.3. |
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10 Dec |
Lecture 14 |
| Reserve lecture. Repetition. |
Section: | |
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