MAT-61756 Measure and Integration, 7 cr
Lisätiedot
Moodle is used during the course. The course is lectured every second year.
Suitable for postgraduate studies.
Ei toteuteta lukuvuonna 2016-2017.
Vastuuhenkilö
Sirkka-Liisa Eriksson
Opetus
| Toteutuskerta | Periodi | Vastuuhenkilö | Suoritusvaatimukset |
| MAT-61756 2016-01 | - |
Sirkka-Liisa Eriksson |
Passed final examination or two partial examinations |
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 applying results in calculations and giving justifications for then. 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. | Lebesgue measure in the set of real numbers starting from the outer measure | An example of the non-measurable set | the role of the axiom of choice in the measure theory |
| 2. | The foundations of the general measure theory: a sigma algebra, a measurable space, a measure space. The theory of integration: Convergence theorems The connections between the Lebesgue integral and the Riemann integral. | Borel measure The connections to the probability theory Ecpected value | Generaöl outer measure A measure given by the general outer measure. |
| 3. | The product measure The integhral with respect to the product measure Tonelli's and Fubini's theorems n-dimensional Lebesgue measure | the Dedekind system the uniqueness theorems the Riemann-Stieltjes integral | |
| 4. | Absolutely continuous measures and singular measures | L-spaces |
Ohjeita opiskelijalle osaamisen tasojen saavuttamiseksi
The grade of the course is based on the final exam or two partial exams. When the points for the final exam or the partial exams are 30% of the maximum, the grade of the course may be improved by bonus points collected from the instructed exercises and homework. The passing limit is 50% of the maximum. If the student is mastering the concepts, results, short proofs and examples type of problems the evaluation is 3. For the grade 4 the student should in addition to the previous level be able to independently apply theory more. For the grade 5 the student should independently deduce results, invent solutions and compare results more than in the previous levels.
Arvosteluasteikko:
Numerical evaluation scale (0-5)
Osasuoritukset:
Oppimateriaali
| Tyyppi | Nimi | Tekijä | ISBN | URL | Lisätiedot | Tenttimateriaali |
| Book | Real analysis | Royden, H.L. | No | |||
| Summary of lectures | Lebesgue measure and integration | S.-L. Eriksson | Available from Moodle | Yes |
Esitietovaatimukset
| Opintojakso | P/S | Selite |
| MAT-01160 Matematiikka 1 | Advisable | |
| MAT-01260 Matematiikka 2 | Advisable | |
| MAT-01360 Matematiikka 3 | Advisable | |
| MAT-01460 Matematiikka 4 | Advisable | |
| MAT-60206 Mathematical Analysis | Advisable | |
| MAT-61256 Geometric Analysis | Advisable |
Vastaavuudet
| Opintojakso | Vastaa opintojaksoa | Selite |
| MAT-61756 Measure and Integration, 7 cr | MAT-61757 Measure and Integration, 5 cr | |
| MAT-61756 Measure and Integration, 7 cr | MAT-41297 Measure and Integral Theory, 8 cr |