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## Syllabus for

TMV100 - Integration theory
Integrationsteori

Owner: MPENM
7,5 Credits
Grading: TH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Education cycle: Second-cycle
Major subject: Mathematics
Department: 11 - MATHEMATICAL SCIENCES

Teaching language: English
Application code: 20134
Open for exchange students: Yes

 Module Credit distribution Examination dates Sp1 Sp2 Sp3 Sp4 Summer course No Sp 0103 Examination 7,5c Grading: TH 7,5c 29 Oct 2020 am J Contact examiner

#### In programs

MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 2 (elective)
MPENM ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 1 (compulsory elective)

Jeffrey Steif

#### Eligibility

General entry requirements for Master's level (second cycle)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

#### Specific entry requirements

English 6 (or by other approved means with the equivalent proficiency level)
Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.

#### Course specific prerequisites

Besides attending a civil engineering program knowledge corresponding to the compulsory courses in mathematics at this program is needed. In addition, a special interest in abstract theories is needed.

#### Aim

The course gives an introduction to the modern theory of integration.

#### Learning outcomes (after completion of the course the student should be able to)

After completion of the course the student should be able to
1. Give motivation to existence of concept of measurability
2. Decide if a collection of sets is a sigma-algebra
3. Prove and apply the Caratheodory's Extension Theorem
4. Explain the concept of measurability and integrability for functions
5. Compaire different types of convergency
6. Prove and apply the Fubini-Tonelli theorem
7. Connect Measure theory and Probability theory
8. Compaire pairs of measures
9. Generalize classical theorems of Analysis to the class of Lebesgue integrable functions/ functons of bounded variation

#### Content

• Measurability
• Integration with respect to a measure
• Lebesgue integral
• Convergence in measure, a.e., and L1
• orthogonality and continuity of measures, Lesbegue-Radon-Nikodym decomposition
• product measure, Fubini-Tonelli theorem
• connection with Probability theory (Borel-Cantelli theorems and Kolmogorov low)
• Lesbegue differentiation
• functions of bounded variation and the Fundamental Theorem of Calculus

#### Organisation

The course ranges over 50 lecture hours; the total effort is about 200 hours.

#### Literature

G. B. Folland: Real Analysis; Modern Techniques and their Applications, John Wiley & Sons.
Ch. Borell: Lecture Notes in MeasureTheory. Department of Mathematics, Chalmers University of Technology and Goteborg University

#### Examination including compulsory elements

A test will be given by the end of the course. A student failing in the ordinary test will be offered additional tests. Hand-in assignments or small projects can be part of the examination. More information about the examination is given on the webb-side of the course before the start of the course.

Page manager Published: Thu 04 Feb 2021.