MS-E1280: Measure and Integral
| Course name | Measure and Integral |
|---|---|
| Course code | MS-E1280 |
| Abbreviation | |
| Period | II |
| Lecturer | Juha Kinnunen |
Description
The course is very abstract and the measure is precisely defined.
After this course a student will know basic methods in measure and integration theory.
Outer measure (properties of measurable sets, characterizations of measurable sets, Lebesgue outer measure), measurable functions (properties of measurable functions, approximation by simple functions, Egoroff and Lusin theorems), integration (construction and properties of integral, Lebesgue integral, convergence theorems), Fubini’s theorem.
Course material
Official material
Lecture notes (PDF, \LaTeX) exist.
Extra material
There are good Finnish lecture notes used at the University of Helsinki that are good to read before the course and during the first weeks.
Contents and workload
As other standard master’s level math courses, this one needs a lot of work. The workload is fairly balanced between weeks.
Note that if this is one of your first master’s courses the workload is higher than in bachelor’s level.
Overall workload
Weekly contents
| Week | Topics |
|---|---|
| 1 | Outer measures, Lebesgue outer measure, measurable sets, $\sigma$-algebras, Carathéodory measurability |
| 2 | Measures, complete and $\sigma$-finite measures, continuity properties of measures, distance function, characterizations of measurable sets |
| 3 | Metric outer measures, Borel outer measures, Lebesgue measure, invariance properties of Lebesgue measure, Lebesgue measurable sets, nonmeasurable sets and the Cantor set |
| 4 | Measurable functions, calculus with infinities, Cantor–Lebesgue function, limits of measurable functions, almost everywhere properties, simple functions |
| 5 | Modes of convergence, Egoroff’s theorem, Lusin’s theorem, integral of nonnegative simple and measurable functions, Monotone convergence theorem, Fatou’s lemma |
| 6 | Signed integrable functions, Dominated convergence theorem, Lebesgue integral, Cavalieri’s principle, Lebesgue and Riemann integrals, Fubini’s theorem |
Practicalities
The course is graded based on homework and there is no exam. The grading is shown below where the percentage means solved homework assignments:
- 90% - 5
- 80% - 4
- 70% - 3
- 60% - 2
- 50% - 1
Related courses
Official prerequisites
MS-A00XX, MS-A01XX, MS-A02XX, MS-A03XX, MS-A050X, MS-C1540.
Additional prerequisites
It is recommended to study the basic properties of measure beforehand. For example, from the University of Helsinki material in Finnish.
More like this
Important in many master’s level math courses. Usually, the basic knowledge of measures is assumed in those even when not mentioned.
MS-E1600: Probability Theory
Trivia
Timestamp
Based on the 2024 version of the course, taught in period II.