Part I, Integration theory

Lectures 1-7 ( Tuesdays): Kristian Bjerklöv. 

Exercise sessions: (Fridays): Luca Sodomaco

 

Topics:

Basics of measure theory

Integration on measure spaces (Lebesgue integral)

Convergence theorems

Product measures, and Fubini's theorem

 Sections 1.1-1.6  (1.7) ;  2.1-2.11, 2.14-2.16 (in Friedman's book)

 

Preliminary planning of the lectures:

Lecture 1: Chapters 1.1-1.2. Recommended problems: 1.1: 2,3,4,6,7; 1.2: 2,3,6.

Lecture 2: Chapters 1.3-1.6. Recommended problems: 1.3: 1,2; 1.4: 4; 1.6: 3,4,5. Here is a note on intervals. Download note on intervals.

Lecture 3: Chapters 2.1-2.4. Recommended problems: 2.1: 6,9,10; 2.2: 2,3; 2.3: 2; 2.4: 3.

Lecture 4: Chapters 2.5-2.7. Recommended problems: 2.5: 2; 2.6: 3,4; 2.7: 3. Here is a note on sigma-finiteness. Download note on sigma-finiteness.

Lecture 5: Chapters 2.8-2.11. Recommended problems: 2.8: 1; 2.9: 1; 2.10: 2,11,12,14.

Lecture 6: Application: Ergodic theory. Short note. Download Short note.

Lecture 7: Chapters 2.14-2.16. Recommended problems: 2.14: 8; 2.16: 1,2,5

 

Homework assignments

There will be two sets of homework assignments, each passed one giving 1 p for the final exam (to part A). The assignments will be posted under Assignments in the left column (one week before the due date).

Assignment 1: Due September 19.

Assignment 2: Due October 10.

The assessment of the homework assignements is as follows. On each (of the two) homework assignment ONE of the problems is randomly chosen. This (and only this) problem is graded (the same problem for all students). If this problem is correctly solved, with a complete solution, one gets 1p.

Solutions to the problems will be presented/discussed at the exercise sessions.