Integration Theory

News

24/1 We covered 1.1 and 1.2. Recommended exercises for 28/1: page 7 number 1, 3, 4, 7 (hard), 8, page 13 number 1, 2, 5, 6, 7, 9 (hard).

31/1 We basically covered everything up until 1.3.3 + 1.4.7, although I just worked with d=1. The case d>1 is conceptually the same and I leave it to you to read. The exercises are 1,2,3 and 5 (hard),7(hard) in section 1.3. I also told you to do a proof. A very similar proof is 1.3.5, which you are supposed to know next time.

4/2 We covered Lebesgue measure in several variables and its basic properties like translation invariance. We also discussed completeness and regularity. In summary, the parts from Chapter 1 which are included in the course is all of 1.1-1.4 except 1.3.7, 1.3.8, 1.4.8 and 1.4.9. The construction of Cantor sets 1.4.6 as well as the entire Section 1.5 are included in the course, but can be omitted if you are only aiming at G. Section 1.6 can be skipped by all.

7/2 We did all of 2.1 and the definition of the integral, which is scattered across 2.3 in the book. Exercises for monday are Sec. 1.4: 1,3 Sec 1.5: 4 Sec 2.1: 2,3,5 and 2.3: 1,2.

14/2 We finished 2.3 and 2.4. The stuff on almost everywhere in 2.2 will be discussed on monday. See pdf file for exercises that I handed out during class. Answers will be handed out on monday. From now on, we will spend more time going through theory on mondays, and instead answers to selected exercises will be handed out. I plan to continue give out my own exercises, which usually contain some silly errors. For example, in Exercise 1 the integral in b) should be over (1,infinity) instead of (0,infinity)... Recommended exercises in the book are Section 2.4 Exercise 2-4, which develops an alternative proof of the DCT (the most important theorem of the course), as well as 8, 9 which contain some "real" applications of the material, similar to the exercises I handed out.

Exercises 1

We have now covered everything in 2.1-2.4. Partial solutions to the exercises are given in the pdf-file.

Answers 1

21/2 We finished chapter 2 and started 3.1. Exercises for 25/2 are Sec 2.2: 6 Sec 2.5: 1,5, Sec 2.6: 2 and all exercises on 3.1. Moreover I have made my own exercise set which is attached as pdf

Exercises 2
Answers to some exercises in 3.1
Answers 2

28/2 We finished 3.2 and 3.3. Recommended exercises in the book are all of 3.2 and from 3.3: 1, 2, 3, 4, 7, 8, 9.

Exercises 3

4/3 We have now finished everything until (but not including) Proposition 3.4.2. We leave the remaining Lp theory to the continuation course. The remaining lectures will go through chapter 5 on partial integration. Attached you find the solution to exercise 3.3.7, as well as solutions to some of the hand out problems.

None

4/3 We have now finished everything until (but not including) Proposition 3.4.2. We leave the remaining Lp theory to the continuation course. The remaining lectures will go through chapter 5 on partial integration. Attached you find the solution to exercise 3.3.7, as well as solutions to some of the hand out problems.

Answer to 3.3.7
Answers 3

7/3 We did all of 5.1 except the proof of 5.3. We also introduced Dynkin classes in 1.6 Next time we will study Dynkin classes and finish sections 5.1 and 5.2. New exercises are found below.

Exercises 4

11/3 We covered products of the Borel and Lebesgue sigma algebras, and how higher dimensional Lebesgue measures arise as products of lower dimensional ones.

Answers 4

Below is a bunch of old exams. Since the older course set up was different, you have to omit some exercises....

From May 2012 you can do 1 a and b, 2, 3 b and 4

May 2012
Solutions may 2012

From Aug 2012 you can do 1a, 2, 3 and 5

August 2012

From May 2010 you can do 1a, 2, 3, 5, 6

May 2010

Here you can do all!

Aug 2010

14/3 We finished all of chapter 5, except Proposition 5.3.1 which contains material that will be covered in the continuation course. Suitable exercises from the book are in Section 5.3, number 1,2,3,6 and 7. On monday and thursday the class will be given by Yacin Ameur. I have instructed him to go through the above exercises and the exam Aug 2012 on monday, and to go through May and Aug 2010 on thursday. You may of course ask him any questions you have before the exam (during class hours...)

List of key concepts and theorems

To prepare for the exams, here is a list of the most important theorems and ideas. The rest is also important and the below is just a recommendation. For VG all what we have covered is of importance. The numbers with * are extra important.

0. Construction of the Borel sigma algebra.
1. Prop 1.2.3
2. Construction of Lebesgue measure via outer measures.
3. Prop 1.4.4 (Leb. meas. is unique translation invariant...)
4. Prop 1.5.6 (Regularity of finite Borel measures. This is a useful fact which we did not show in class, so its only for VG aspirants.)
5. Facts regarding differences concerning Borel and Lebesgue sigma algebras. (All are supposed to be aware of these differences, but proofs are for VG aspirants.)
6. Construction of the Lebesgue integral.
7. Prop 2.1.7 (how to approximate measurable functions with simple ones), Prop 2.3.8 (null sets can be ignored in integrals), Cor 2.3.12 (integrable functions are finite a.e.)
8. Monotone convergence theorem, Dominated convergence theorem, Fatou's lemma.
9. Thm 2.5.1 (Riemann integral=Lebesgue integral when it exists...)
10. Integration of complexvalued functions, Prop 2.6.4
11. Section 3.1
12. Hölder and Minkowski inequalities
13. Construction of Lp spaces and their completeness, understand why and how we replace functions with equivalence-classes
14. Fubini's theorem

Exams

The exam is in 309b on friday the 22/3 between 10.15 and 15.15.

Corrected exams will be shown at 12.30-13.15 in 332 A on the 8th of april. If you need to know your result sooner or can not show up, write me an email mc@maths.lth.se

For those who pass the written exam there will be an oral exam, to be taken during april. This can take between 30 minutes and 2 hours, depending on your level and whether you want VG. Appointments are scheduled on the 8th or by sending me an email.

Last exam with solutions

Exam March 2013

Course Program

Lectures: Mondays and Thursdays, 13:15-15:00 in room 332A

Literature

Exams