Study Guide 2015-2016

MAT-61756 Measure and Integration, 7 cr

Additional information

Moodle is used during the course. The course is lectured every second year.
Suitable for postgraduate studies

Person responsible

Sirkka-Liisa Eriksson

Lessons

Implementation 1: MAT-61756 2015-01

Study type P1 P2 P3 P4 Summer
Lectures
Excercises
  4 h/week
 3 h/week
 +3 h/week
+3 h/week

 

 

 

Lecture times and places: Monday 10 - 12 TB215 , Tuesday 10 - 12 TB215 , Thursday 12 - 15 TC417 , Thursday 12 - 15 TB206 , Monday 12 - 15 TB220 , Thursday 12 - 15 TB216

Requirements

Passed final examination or two partial examinations
Completion parts must belong to the same implementation

Learning Outcomes

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.

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 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   

Instructions for students on how to achieve the learning outcomes

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:

Completion parts must belong to the same implementation

Study material

Type Name Author ISBN URL Additional information Examination material
Book   Real analysis   Royden, H.L.         No   
Summary of lectures   Lebesgue measure and integration   S.-L. Eriksson       Available from Moodle   Yes   

Prerequisites

Course Mandatory/Advisable Description
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    



Correspondence of content

Course Corresponds course  Description 
MAT-61756 Measure and Integration, 7 cr MAT-41297 Measure and Integral Theory, 8 cr  

Last modified 27.03.2015