Exercise 1.3 in Section 1 of the appendix states properties of measures
that I quoted in the first lecture.

The theorem on equivalence of measures and distribution functions that
I stated in the second lecture but did not prove is Theorem 1.5 of the 
Appendix.  

After that, comes a discussion of distribution functions in more than 
one dimension -- I think this is only important for a couple of proofs 
of lemmas and then you never see it again...

In the section on Cartheodory's extension theorem, there is one result
that is important for proving that something is true for all sets in
a given sigma-field (Math 546 students need to know this): the so-called
Pi-Lambda Theorem (2.1).  

Section 3 of the appendix is not relevant.  Some of you may have seen
measure theory developed for the real numbers, e.g. in Rudin's 
"Principles of Mathematical Analysis" (think blue book).  To clear
up any confusion, here are the differences between that approach and
ours.  We define "measurable" to mean a set is in our favorite
sigma-field (for the reals, the Borel sigma-field, or perhaps the 
Lebesgue sigma-field).  For us it is a (easy) theorem that for
all Borel sets, inner measure equals outer measure.  We then complete
the Borel sigma-field to the Lebesgue sigma-field, where again it
is easy to show inner measure equals outer measure.  Rudin and others 
define "measurable" as "inner measure equal to outer measure", and 
then prove that this includes all Borel sets.

For the details on the construction of the Lebesgue integral in the
third lecture, read Section 4 of the appendix.  

The definition of a distribution having a density, given at the top of
page 6, is the one which I gave in class after being asked for clarification.  
Note that what is required is for F to have a derivative, f, "in the
sense of distributions", meaning you need F = integral of f, not 
f = derivative of F.  In fact the notion of the "derivative" of F, makes
sense only if we're talking about distributions on the reals, while
the notion "F = integral of f" makes sense on any abstract space.
The Fat Cantor set example shows that one can have F = integral of f,
but F nowhere differentiable in the classical sense.