Comments on Assignment 2

Problem 1: Mainly think about (a). In (b), you
could comment how your proof in (a) uses countable
additivity. One can construct examples of finitely 
additive measures that satisfy the assumptions but are not
delta-measures, using tools of Set Theory (ultrafilters) 
or Functional Analysis (the Hahn-Banach theorem), but those
are beyond the scope of this course.

Problem 2: This is the Vitali-Hahn-Sachs theorem.
Its proof is elementary, but quite involved. I will 
post a solution on Wednesday.  (Full credit to all).

Problem 3:
A standard tool in Probability.

Problem 4:
You may replace this by Problem 14 of Folland
(which requires a similar, but easier argument).
... if you've already solved Problem 4, more power to you ...

Problem 5: Let L be the union of all sets in the chain.
Consider a countable dense subset {x_n} of L,
then pick F_n in the chain such that F_n contains x_n. 
If the union of the F_n's is a proper subset of L,
pick a point y in the difference, and let E be a set 
in the chain containing y. How does E relate to the 
F_n's?  ... and don't forget that E is closed ...

Tha argument sketched above works for any separable 
space in place of the real line.  One can get by 
with a bit less, using only that each point has a 
countable neighborhood base. In that argument,
take a point x in the closure of L, and let F_n
be a set n the chain that intersects the ball
of radius 1/n about x.