MAT-61757 Measure and Integration, 5 cr
Lisätiedot
Suitable for postgraduate studies.
Vastuuhenkilö
Janne Kauhanen
Opetus
| Toteutuskerta | Periodi | Vastuuhenkilö | Suoritusvaatimukset |
| MAT-61757 2017-01 | 4 |
Janne Kauhanen |
Final exam, weekly exercises, and weekly written exercises. |
Osaamistavoitteet
After the completion of the course the student knows the main concepts and results of the measure and integration theory. The student is capable of defining the main concepts precisely. The student is capable of writing justifications and applying results in simple proofs and calculations. The student is able to verify the most important results. The student can apply the concepts and results in advanced studies and applications in the area of analysis and stochastics. The exact mathematical reasoning is emphasized during the course.
Sisältö
| Sisältö | Ydinsisältö | Täydentävä tietämys | Erityistietämys |
| 1. | Unions and intersections with general index sets, countable and uncountable sets. The limit superior and the limit inferior. Sigma algebra and measure. Outer measure and measurable sets. | The Axiom of Choice | |
| 2. | Lebesgue measure in the set of real numbers starting from the outer measure. Borel sets and Lebesgue measurable sets. | Borel measure. An example of a non-measurable set. | General outer measure. The measure given by a general outer measure. Probability measure. |
| 3. | Lebesgue measurable functions. The Lebesgue integral: the integral of a simple function, the integral of a nonnegative function and the Monotone Convergence Theorem, the integral of a measurable function and the Lebesgue Dominated Convergence Theorem. | The connections between the Lebesgue integral and the Riemann integral. | L^p spaces. The Lebesgue integral in R^n and Fubini's Theorem. |
Oppimateriaali
| Tyyppi | Nimi | Tekijä | ISBN | URL | Lisätiedot | Tenttimateriaali |
| Book | Measure, Integration & Real Analysis (Preliminary Edition) | Axler, Sheldon | No |
Esitietovaatimukset
| Opintojakso | P/S | Selite |
| MAT-60206 Mathematical Analysis | Mandatory |
Vastaavuudet
| Opintojakso | Vastaa opintojaksoa | Selite |
| MAT-61757 Measure and Integration, 5 cr | MAT-61756 Measure and Integration, 7 cr |