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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)
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 Fall '10
 GREGORYMARGULIS
 Algebra, Sets

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