Math 447
Solutions to Assignment 2
Spring 2009
Exercise 16.22 on page 273
Let
E
=
S
∞
n
=1
E
n
. Show that
m
*
(
E
) = 0 if and only if
m
*
(
E
n
) = 0 for every
n
.
Solution
Since
E
n
⊆
E
, the monotonicity property of outer measure [part (ii) of Propo
sition 16.2 on page 269] shows that
m
*
(
E
n
)
≤
m
*
(
E
). If the righthand side
m
*
(
E
) = 0,
then
m
*
(
E
n
) = 0 for every
n
. In other words, a subset of a set of measure zero is again a
set of measure zero.
On the other hand, the countable subadditivity of outer measure (Proposition 16.5 on
page 272) implies that
m
*
(
E
)
≤
∑
∞
n
=1
m
*
(
E
n
). If
m
*
(
E
n
) = 0 for every
n
, then the
righthand side equals 0, so
m
*
(
E
) = 0. (This part of the exercise is Corollary 16.6 on
page 272.)
Exercise 16.24 on page 273
Given a subset
E
of
R
, prove that there is a
G
δ
set
G
containing
E
such that
m
*
(
G
) =
m
*
(
E
).
Solution
Recall from page 130 that a
G
δ
set means the intersection of a countable number
of open sets.
By the deﬁnition of outer measure, there exists for each positive integer
n
an open set
G
n
containing
E
such that
m
*
(
G
n
)
≤
m
*
(
E
) + 2

n
. Set
G
equal to the intersection
T
∞
n
=1
G
n
.
Then
G
is a set of type
G
δ
. Moreover,
E
⊆
G
n
for every
n
, so
E
⊆
G
. Since
G
⊆
G
n
for every
n
, it follows that
m
*
(
G
)
≤
m
*
(
E
) + 2

n
for every
n
, so
m
*
(
G
)
≤
m
*
(
E
). By
monotonicity of outer measure,
m
*
(
E
)
≤
m
*
(
G
). Combining the preceding two inequalities
shows that
m
*
(
G
) =
m
*
(
E
). Thus
G
has all the required properties.
Exercise 16.28 on page 273
Fix
α
with 0
< α <
1 and repeat our “middle thirds”
construction for the Cantor set except that now, at the
n
th stage, each of the 2
n

1
open
intervals we discard from [0
,
1] is to have length (1

α
)3

n
. (We still want to remove each
open interval from the “middle” of a closed interval in the current level — it is important
that the closed intervals that remain turn out to be nested.) The limit of this process, a
set that we will name Δ
α
, is called a
generalized Cantor set
and is very much like the
ordinary Cantor set. Note that Δ
α
is uncountable, compact, nowhere dense, and so on,
but has nonzero outer measure. Indeed, check that
m
*
(Δ