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Course Catalog 2012-2013
MAT-41297 Measure and Integral Theory, 8 cr |
Additional information
The course is the English version of the Finnish course MAT-41291. The language of the lectures depends on how many English speaking students are attending. Homework and instructed exercises are in English
Suitable for postgraduate studies
Will not be lectured year 2012-2013
Person responsible
Sirkka-Liisa Eriksson
Principles and baselines related to teaching and learning
During the course there are instructed exercises that helps learning. Moodle study platform is used during the course and extra material may be handed out through that there and students may discuss about there problem.
Learning outcomes
After the completion of the course the student knows the main concepts anf 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 stocastics. The exact mathematical reasoning is emphesized during the course.
Content
| Content | Core content | Complementary knowledge | Specialist knowledge |
| 1. | Lebesgue measure in the set of real numbers starting from the outer measure | An example of the mnon-measurable set | Applying the axiom of choice in the measure theory |
| 2. | The foundations of the general measure theort: a sigma algebra, a measure, a measure spacem. Integrointiteoriaa. Konvergenve theorems The connections between the Lebesgue integral and teh 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 theoremst the Riemann-Stieltjes integrl | |
| 4. | Absolutely continuous measures and singular measures | L-spaces |
Evaluation criteria for the course
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.
Assessment scale:
Numerical evaluation scale (1-5) will be used on the course
Partial passing:
Prerequisites
| Course | Mandatory/Advisable | Description |
| MAT-43850 Mathematical Analysis 2 | Advisable |
Prerequisite relations (Requires logging in to POP)
Correspondence of content
| Course | Corresponds course | Description |
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