Homework 1 due on Monday October 12, 98
1. Let 
be a measurable set
and 
be a function on 
with values in a set X. Prove that
is a 
-field.
2. Show that a subset 
of a Polish space
is a compact set if and only if it is closed
and totally bounded.
3. Let 
be a family of finite measures on a Polish space 
and let 
be a finite measure on .
Prove that if any sequence of elements
of 
has a subsequence weakly convergent to ,
then 
.
4. Let 
be an irrational number.
For 
, denote by 
the fractional part of 
. Prove that
for any 
5. Prove that the family of all finite dimensional
cylindrical sets is an algebra. (Hint: attached points 
and 
may vary.)
6. Let 
denote the cylindrical -field
in the set of all 
-valued functions on 
. Prove that for
any
there exists a countable set 
such that if 
and 
is a function
such that 
for all , then 
.
In other words, elements of are defined by specifying
conditions on trajectories only at countably many
points of .
7. Give an example of a Polish space such that
the set 
of all bounded and continuous -valued
functions
on is not an element of the -field
from previous exercise. Thus you will see that there exists
a very important and
natural set which is not measurable.