COMMENTS TO ASSIGNMENT 1

PROBLEM 1

Corrections: 
Typo in line 1: R is a subset (not an element) of P(X).
Also, we should require that R is non-empty.

The ring generated by C is actually more complicated
than what I wrote in Part (b).
I've reverted to the tested-and-true problem from Folland. 
Please look at the corrected assignment.
The original question will re-appear -- corrected -- later.

Comments:
Parts (c) and (d) will be used in the proof of Caratheodory's
extension theorem


PROBLEM 2

Notation: 
If A is a set, the symbol "#A" means the number of elements 
(or its cardinality, if the set is infinite).

This idea of "density" is sometimes used to motivate 
statements in Number Theory probabilistically.  
Note that the density is additive, i.e., 

   density (E union F) = density (E) + density (F)

for disjoint sets E and F, provided that both lie in C. 
But the density is not a finitely additive measure. (Why?)


PROBLEM 3

Hints:
(a) Starting from a sequence of distinct sets in M, 
construct a sequence of non-empty disjoint sets.
Apply intersections and complements ...

(b) Try to construct for each binary sequence a 
different element of M.


PROBLEM 4

This fact is used in the proof of the Monotone Class Theorem.


PROBLEM 6:

Notice that the liminf is the union of an increasing
sequence of sets.