1.9
Exercises
25
31.
Suppose
n
is uncountable and let
g
be the a-field consisting
of
sets
A
such
that either
A
is countable or
A
c
is countable. Show
g
is NOT countably
generated. (Hint:
If
9
were countably generated, it would be generated by
a countable collection
of
one point sets. )
In fact,
if
g
is the a-field
of
subsets
of
n
consisting
of
the countable and
co-countable sets,
g
is countably generated iff
Q
is countable.
32. Suppose B1,
B2
are
a
-fields
of
subsets
of
n
such that
B1
c
Bz
and
B2
is
countably generated. Show by example that it is not necessarily true that
B1
is countably generated.
3 3.
The extended real line.
Let
i
=
lR
U
{-
oo}
U {
oo}
be the
extended
or
closed
real line with the points
-oo
and
oo
added. The Borel sets
B(JR)
is the a-field generated by the sets
[-oo,x],x
E
JR,
where
[-oo,x]
=
{
-oo}U(
-oo,
x]
.
Show
B(JR)
is also generated by the following collections
of
sets:
(i) [
-00,
X), X
E
JR,
(ii)
(x,
oo],
x
E
JR,
(ii)
all finite intervals and {
-oo}
and {
oo}
.
Now think
of
i
= [
-oo,
oo]
as homeomorphic in the topological sense to
[
-1,
1]
under the transformation
X
X
t-+
--
1-lxl
from [
-1,
1]
to [
-oo,
oo].
(This transformation is designed to stretch the
finite interval onto the infinite interval.) Consider the usual topology on
[
-1,
1]
and map it onto a topology on [
-oo,
oo]
.
This defines a collection
of
open sets on [
-oo,
oo]
and these open sets can be used to generate a
Borel a-field. How does this a-field compare with
B(JR)
described above?
34. Suppose
B
is a a-field
of
subsets
of
n
and suppose
A
¢
B.
Show that
a(B
U {A}), the smallest a-field containing both
Band
A
consists
of
sets
of
the form
ABu
ACB',
B,
B'
E
B.
35. A a-field cannot be countably infinite. Its cardinality is either finite
or
at
least that
of
the continuum.
36. Let
n
=
{f,
a, n,
g}, and
C
=
{{f,
a, n},
{a,
n}}.
Find
a(C).
37. Suppose
n
=
Z, the natural numbers. Define for integer
k
kZ
=
{kz
:
z
E
Z}.
Find
B(C)
when
C
is