COMMENTS TO ASSIGNMENT 2


PROBLEM 1

Hint: Subdivide and conquer.
(Notice that the definition of \delta_a has been added.)

Gratuitous question: What if \mu is only finitely additive?


PROBLEM 2

Use the Cauchy criterion.


PROBLEM 3

(a) Induction over n will work. Note that the sum runs over
all non-empty subsets of {1, ..., n}.
(There is a slicker argument that uses indicator functions.)

(b) Let A_i = { permutations in S_n that fix i } and include-exclude.

Small comment on notation: I am thinking of a permutation \pi as 
a map from {1, ..., n} to itself.  If you visualize the permutation
as a string of n numbers, then \pi(i) is the i-th one.


PROBLEM 4

Consider the collection C of sets in the plane
whose cross section at a given height y is Borel. 

(What can you say about the cross section A(y) if 
A is an open set?  A picture may help ...)


PROBLEM 5

Hint: Write Lambda' as a union of finite sets. 


PROBLEM 6

The difficulty is that the collection C may be uncountable.
But note that the union (as a subset of the real line) is separable, 
i.e., it contains a countable dense subset S = { x_n,  n=1,2, ... }. 
Use this to replace C by a (countable) sequence.