Math 320 Measure Theory and Integration
Assignment 1:
&
algebras and Borel sets
The due date for this assignment is Thursday 9/9/2010.
1. Let
A
be the set of numbers in
[0
;
1]
which admit decimal expansions such that the
digits
2
;
4
;
6
;
8
all appear at least once. (Thus, 0.0032004068 is an example while
0.232046 is not. For 0.9999.
..., write 1 instead.) Show that
A
is a Borel set.
Solution.
Let
E
i
stand for the set of points in
[0
;
1]
such that the digit
1
< i <
9
appears in the decimal expansions of the points. Let
F
(
i
)
j
denote the set of numbers
with
i
appearing in the
j
th decimal place. We have
F
(
i
)
j
=
[
10
j
1
k
=0
[
k
10
j
+
i
10
j
;
k
10
j
+
i
+1
10
j
)
.
Then
E
i
=
[
1
j
=1
F
(
i
)
j
. Note that
E
i
is a Borel set. Finally, the set we want to describe
is
\
i
=2
;
4
;
6
;
8
E
i
with each
E
i
a Borel set. Hence, it is Borel.
2. Given sets
X
and
Y
. Let
f
:
X
!
Y
be any function into
Y
.
(a)
Show that if
A
is a
&
algebra of subsets of
Y
, then
f
1
(
A
)
is a
&
algebra of
subsets of
X
.
(b)
Show that
(
f
1
(
S
)) =
f
1
(
(
S
))
, for any collection
S
of subsets of
Y
. Here
(
K
)
represents the
&
algebra generated by
K
.
Solution.
(a)
[1]
X
2
f
1
(
A
)
since
Y
2 A
and
f
(
X
) =
Y
. [2] Note we have
f
1
(
A
c
) =
(
f
1
(
A
))
c
for any
A
. (In general it is
not
true that
f
(
A
c
) =
f
(
A
)
c
.) We have
A
2
f
1
(
A
)
()
A
=
f
1
(
B
)
for some
B
2 A
:
Thus
A
c
= (
f
1
(
B
))
c
=
f
1
(
B
c
)
2
f
1
(
A
)
(since
B
2 A
). [3] Let
A
1
;A
2
;:::
2
f
1
(
A
)
. We want to
show
[
A
i
2
f
1
(
A
)
. We have
A
i
=
f
1
(
B
i
)
for
i
= 1
;
2
;
3
:::
with
B
i
2 A
. Now,
[
A
i
=
[
f
1
(
B
i
) =
f
1
(
[
B
i
)
2
f
1
(
A
)
, since
[
B
i
2 A
.
(b)
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 GREGORYMARGULIS
 Algebra, Topology, Sets, Empty set, Metric space, Open set, Topological space, Borel set

Click to edit the document details