# Integration Theory

In this course we will cover most parts of chapters 1-3 and 5 in Cohn's book. In the continuation course, MATP29, we will cover the remaining parts of those chapters plus chapters 4 and 6 and some parts of chapter 7.

Note that the course book is available as an e-book through Lund University (follow the link).

Summary of Lectures:

• Lecture 1, Jan 22:

We discussed the introduction and chapters 1.1-1.2. The discussion of G_delta and F_sigma in Ch. 1.1 is not that important. You can skip it and Prop. 1.1.6. You can also skip Prop 1.1.7. You should read everything else. The most important concepts are of course sigma-algebras and measures.

• Lecture 2, Jan 26:

We covered most of Ch. 1.3 on outer measures (until and including Thm. 1.3.6). The most important parts are of course the definition of outer measures and the construction of a sigma-algebra and a measure from an outer measure, the chief example being the Lebesgue measure. We will discuss Prop. 1.3.7 and 1.3.8 on Thursday. You should have a look at pages 19-21, but you don't have to learn all the details. The measures discussed there are a special case of Lebesgue-Stieltjes measures.

• Lecture 3, Jan 29:

We finished Ch. 1.3 (Prop. 1.3.7 and 1.3.8 + a summary of Prop. 1.3.9 and 1.3.10 without proofs). We also talked about Prop. 1.4.1 in Ch. 1.4. We have now discussed the exercises in Ch. 1.1 and 1.2. We're a bit behind according to the original schedule. I've therefore posted an updated version below.

• Lecture 4, Feb 2:

We finished Ch. 1.4. I only discussed the first construction of a set which is not Lebesgue measurable (Thm 1.4.9). If you want to know more, you can read about a stronger result in Prop. 1.4.11. We also discussed exercises 1, 2 and 3 from Ch. 1.3. We'll discuss ex. 5 and 7 on Thursday.

• Lecture 5, Feb 5:

We discussed Ch. 1.5. I skipped some of the proofs. I won't require the proof of Prop. 1.5.6 (and Lemma 1.5.7) for the oral exam, but you should read the rest of the proofs. We also discussed exercises 5 and 7 from Ch. 1.3 and exercise 3 from Ch. 1.4. I've posted solutions to ex. 1, 4 and 5 in Ch. 1.4 below. Note that these are only suggestions. There are many different ways to solve the exercises.

• Lecture 6, Feb 10:

We discussed Ch. 2.1 and 2.2. I presented the material in a slightly different way. I defined a real-valued function to be measurable if the preimage of any Borel set is measurable. Similarly, an extended real-valued function is measurable if the preimage of any Borel subset of the extended real line is measurable, where the Borel sigma-algebra on the extended real line consists of sets whose intersection with the real numbers is a usual Borel set. In other words, any such set is either a usual Borel set or a union of a usual Borel set with {-infinity} and/or {+infinity}. These definitions are completely equivalent to the definition in the book. Fortunately, it suffices to check a collection of sets which generates the Borel sigma-algebra. I skipped Example 2.10 and Proposition 2.1.11. We will get back to this in the continuation course, but you should at least know that there are Lebesgue measurable sets which are not Borel measurable. In Ch. 2.2 I discussed everything except Prop. 2.2.5, which I will let you read on your own. We also discussed ex. 1 in Ch. 1.5. We'll try to cover the remaining exercises on Thursday. Ex. 9 in Ch. 1.5 is more challenging. A hint is to use ex. 5 and to first consider the case of a finite measure.

• Lecture 7, Feb 12:

We discussed the construction of the integral in Ch. 2.3. The last result we discussed was Prop. 2.3.9. We'll discuss the remaining parts on Monday. Everything in this chapter is included in the course. We also discussed exercises 2, 3 and 5 in Ch. 2.1. Since we didn't have time to discuss the remaining exercises in Ch. 1.5 and exercise 6 in Ch. 2.2, I've posted solutions below.

• Lecture 8, Feb 16:

We finished Ch. 2.3 and then discussed Ch 2.4. This is one of the most important chapters in the book and everything in it is included in the course. These limit theorems are among the key reasons for using the Lebesgue integral instead of the Riemann integral. I will give some more concrete examples of applications on Thursday. We also discussed the exercises in Ch 2.3.

• Lecture 9, Feb 19:

We discussed Ch. 2.5. The part about Riemann sums on pp. 70-71 is left for self-study (I assume that you have seen this in previous courses).The main point of this chapter is that the (proper) Riemann integral equals the Lebesgue integral whenever it is defined. This is also true for absolutely convergent improper Riemann integrals. Thus we can use all the usual methods (the fundamental theorem of calculus, integration by parts etc.) to calculate Lebesgue integrals of Riemann integrable functions. We also discussed a concrete example of how to apply the dominated convergence theorem and exercises 2, 3 and 10 in Ch. 2.4. I've posted solutions to exercises 4 and 9 below. You had some questions about when it is allowed to use properties (a)-(d) in Prop. 2.3.6. This proposition is formulated in a restrictive way and most of the results hold more generally. E.g. if f and g are extended real-valued and integrable, then all of the properties still hold (as can be seen using Prop. 2.3.9 and Cor. 2.3.14). The only problem is that sum f+g might not be defined everywhere. It is however defined almost everywhere and it doesn't matter how we define it at the remaining points. I've included solutions to exercises 2 and 3 in the pdf file below, so that you can see how one can argue about these matters. Usually these details are not written out explicitly, but it's good training to write out all the steps in the beginning.

• Lecture 10, Feb 23:

We discussed Ch. 2.6. We've already discussed some of this in connection with the definition of measurable functions. I discussed complex-valued functions in detail and left image measures for self-study. We also discussed Ch. 3.1 which deals with various forms of convergence of sequences of measurable functions. We discussed everything except Prop. 3.1.6 which I'll let you read on your own. I've posted solutions to the exercises we didn't discuss in the lecture below. I've saved problem 4 of the additional exercises for Thursday.

• Lecture 11, Feb 26:

We discussed Ch. 3.2 which concerns normed vector spaces and metric spaces. Everything is included, but I skipped the proof of Prop. 3.2.5 in the lecture. You should read this on your own. We also started on Ch. 3.3. We got as far as proving Lemma 3.3.1. We will do the rest on Monday. Since you need some of the remaining parts of the chapter to solve Exercise 3.3.3, we will defer it to next Thursday.

• Lecture 13, March 2:

We finished Ch. 3.3 (everything is included) and discussed Prop. 3.4.2-3.4.4 in Ch. 3.4. Thm. 3.4.1 will be discussed next time. Prop. 3.4.5 and Lemmas 3.4.6 and 3.4.7 are not included. I've posted solutions to the exercises below (recall that Ex. 3.3.3 will be discussed on Thursday).

• Lecture 14, March 5:

We finished Ch. 3.4 by discussing Thm. 3.4.1. We then started the last part of the course which is about product measures and Fubini's theorem. We discussed the definition of the product of two sigma-algebras in Ch. 5.1 and showed that the product of the Borel sigma-algebra on R with itself is the Borel sigma-algebra on R^2. We then discussed the theory in Ch. 1.6 which can be used to prove uniqueness results. The main idea is that if two measures agree on a collection of sets which generates a sigma-algebra, we can under some additional hypotheses say that the measures must agree on the sigma-algebra. We discussed exercise 7 in Ch. 3.3 and exercise 2 in Ch. 3.4. Solutions to the other exercises can be found below. Note that there is an error in the statement of exercise 7 in Ch. 3.3 (see the solutions).

• I've posted three old exams below. We will discuss the first two (March 24, 2014 and April 20, 2013) in class; you will get written solutions to the third one (March 22, 2013). I've also posted a summary of the course, which might be useful for preparation for the exams.
• Lecture 15, March 9:

We finished Ch. 5.1 and started on Ch. 5.2 (we proved Prop. 5.2.1). Everything in these two chapters is included.

• Lecture 16, March 12:

Alexandru Aleman substituted for me. In Ch 5.3 everything except Prop. 5.3.3 is included.

• Lectures 17 and 18, March 16 and 19:

We discussed the the exercises in Ch. 5.2 and 5.3 and the exams from March 2014 and April 2013. Since all problems from the 2014 exam were not discussed in class, I've posted solutions for the remaining problems below. I've also posted solutions to the exam from March 2013.

• Solutions to the exam can now be found below.

## Course Material

Programme 1
Programme 2
Programme 3
Programme 4
Practice exams
Review for the exams
Solutions to some of the problems in Ch. 1.4.
Solutions to some of the problems in Ch. 1.5
Solutions to some of the problems in Ch. 2.2
Solutions to some of the problems in Ch. 2.4
Solutions to some of the problems in Ch. 2.5 and some of the additional exercises
Solutions to some of the problems in Ch. 2.6 and 3.1
Solutions to some of the problems in Ch. 3.2
Solutions to some of the problems in Ch. 3.3
Solutions to some of the problems in Ch. 3.4
Solutions to some of the problems in Ch. 1.6 (updated March 31)
Solutions to the exam from March 2013
Solutions to the remaining problems on the exam from 2014 (updated March 31)

## Previous exams

2015-03-21 Exam Solutions
2014-08-27 Exam
2014-04-10 Exam
Questions:

## Course Start

Introductory Meeting:
2015-01-19, 16:15
MH: C

Start date:
2015-01-19