Course syllabus for Integration theory

The course syllabus contains changes
See changes

Course syllabus adopted 2019-02-22 by Head of Programme (or corresponding).

Overview

  • Swedish nameIntegrationsteori
  • CodeTMV100
  • Credits7.5 Credits
  • OwnerMPENM
  • Education cycleSecond-cycle
  • Main field of studyMathematics
  • DepartmentMATHEMATICAL SCIENCES
  • GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail

Course round 1

  • Teaching language English
  • Application code 20134
  • Open for exchange studentsYes

Credit distribution

0103 Examination 7.5 c
Grading: TH		7.5 c
  • 29 Okt 2020 am J
  • Contact examiner

In programmes

Examiner

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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.

The course syllabus contains changes

  • Changes to course rounds:
    • 2020-09-05: Examinator Examinator changed from Maria Roginskaya (maria) to Jeffrey Steif (steif) by Viceprefekt
      [Course round 1]
  • Changes to examination:
    • 2021-04-14: Exam date Exam date changed by Elisabeth Eriksson
      [32722, 53795, 3], New exam for academic_year 2020/2021, ordinal 3 (not discontinued course)
    • 2020-09-30: Grade raising No longer grade raising by GRULG