Integration Theory

These lecture notes are written when the course in integration theory is for the first
time in more than twenty years, given jointly by the the two divisions Mathematics
and Mathematical Statistics. The major source is G. B. Folland: Real Analysis,
Modern Techniques and Their Applications. However, the parts on probability
theory are mostly taken from D. Williams: Probability with Martingales. Another
source is Christer Borell’s lecture notes from previous versions of this course, see
www.math.chalmers.se/Math/Grundutb/GU/MMA110/A11/
2 Introduction
This course introduces the concepts of measures, measurable functions and Lebesgue
integrals. The integral used in earlier math courses is the so called Riemann integral.
The Lebesgue integral will turn out to be more powerful in the sense that
it allows us to define integrals of not only Riemann integrable functions, but also
some functions for which the Riemann integral is not defined. Most importantly
however, is that it will allow us to rigorously prove many results for which proofs
of the corresponding results in the Riemann setting are usually never seen by students
at the basic and intermediate level. Such results include precise conditions
for when we can change order of integrals and limits, change order of integration
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in multiple integrals and when we can use integration by parts. Of course, we will
also prove many new results.
The concept of measurability is an advanced one, in the sense that a lot of
people at first find it difficult to master; it tends to feel fundamentally more abstract
than things one has encountered before. Therefore, a natural first question is why
the concept is needed. To answer this, consider the following example.
Let X = R=Z, the circle of circumference 1, with addition and multiplication
defined modulo 1. Suppose we want to introduce the concept of the length of
subsets of X. A natural first assumption is that one should be able to do this so
that the length is defined for all subsets of X. It is also extremely natural to claim