30
SPRING 2012
12.
Lebesgue Measure: Open Sets and Compact Sets
So far we have defined
λ
(
A
) for
A
empty, for any closed box
A
, and more generally for
any special polygon
A
⊆
R
n
.
Definition 12.1.
If
G
⊆
R
n
is a nonempty open set, let
λ
(
G
) = sup
{
λ
(
P
)

P
is a special polygon such that
P
⊆
G
}
.
Since every nonempty open set contains an open ball in
R
n
, and every open ball contains
a nondegenerate closed box, we have 0
<
λ
(
G
) for any nonempty open set
G
.
Examples 12.2.
(i)
λ
(
R
n
) =
∞
. To see this, fix
M >
0 and consider the closed box
I
M
= [0
, M
]
×
[0
,
1]
×
· · ·
×
[0
,
1].
Then
I
M
is a special polygon and
I
M
⊆
R
n
,
λ
(
R
n
)
≥
λ
(
I
M
) =
M
. Since
M >
0 was arbitrary, we have
λ
(
R
n
) =
∞
.
(ii) If
G
is a bounded nonempty set, then
λ
(
G
)
<
∞
. To see this, by definition there
exists
K
∈
R
such that
G
⊆
B
K
(
0
). So if we let
J
K
be the closed box [
−
K, K
]
n
,
then
G
⊆
J
K
. For any special polygon
P
with
P
⊆
G
we thus have
P
⊆
J
K
, and
therefore
λ
(
P
)
≤
λ
(
J
K
) = (2
K
)
n
<
∞
.
Taking the sup over all such
P
’s gives
λ
(
G
)
≤
(2
K
)
n
<
∞
.
Exercise 12.1.
Show that for any special polygon
P
, we have
λ
(
P
) =
λ
(
P
o
), where
P
o
denotes the
interior
of
P
(that is, the largest open set contained in
P
).
Hint:
First consider
a closed box
I
, and argue that for any
>
0, we can construct a closed box
J
such that
J
⊆
I
0
⊆
I
and
λ
(
I
)
−
λ
(
J
)
<
.
Proposition 12.3.
Let
G
i
⊆
R
n
be open for each
i
∈
N
.
(i)
If
G
1
⊆
G
2
, then
λ
(
G
1
)
≤
λ
(
G
2
)
.
(ii)
λ
(
∪
∞
i
=1
G
i
)
≤
∞
i
=1
λ
(
G
i
)
, with equality holding if the
G
i
are
(
pairwise
)
disjoint.
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 Spring '09
 Topology, Sets, Empty set, Metric space, k2, General topology, special polygon

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