Course syllabus for Integration theory

Course syllabus adopted 2024-02-01 by Head of Programme (or corresponding).

Overview

  • Swedish nameIntegrationsteori
  • CodeTMV101
  • 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 20157
  • Open for exchange studentsYes

Credit distribution

0124 Oral examination 7.5 c
Grading: TH		7.5 c

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

Mathematics (at least 37.5 cr. including Linear Algebra, Multivariable Analysis, Mathematical Statistics)

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. 

Jeffrey Steif: Lecture Notes on measure theory and integration theory as well as differentiation theory

Examination including compulsory elements

Oral examination

The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers on educational support due to disability.