ASSIGNMENT 8
·
SOLUTIONS
MAT 473
·
SPRING 2012
Exercise 14.1.
For
A
⊆
R
n
and
x
∈
R
n
, prove that
A
+
x
∈
L
if and only if
A
∈
L
, in
which case
λ
(
A
+
x
) =
λ
(
A
).
Proof.
For any
x
∈
R
n
, translation by
x
is continuous on
R
n
with continuous inverse (namely,
translation by
−
x
). It follows that a set
G
is open if and only if
G
+
x
is open. Also, recall
from Exercise 12.2 that
λ
(
G
+
x
) =
λ
(
G
) for all open
G
⊆
R
n
.
Now consider any set
A
⊆
R
n
. Since a set
G
is open with
A
+
x
⊆
G
if and only if
G
−
x
is open with
A
⊆
G
−
x
, it follows that
λ
∗
(
A
+
x
) = inf
{
λ
(
G
)

G
is open and
A
+
x
⊆
G
}
= inf
{
λ
(
G
)

G
−
x
is open and
A
⊆
G
−
x
}
= inf
{
λ
(
G
−
x
)

G
−
x
is open and
A
⊆
G
−
x
}
= inf
{
λ
(
H
)

H
is open and
A
⊆
H
}
=
λ
∗
(
A
)
.
Since a set
K
is compact if and only if
K
+
x
is compact, it follows from this that
λ
(
K
+
x
) =
λ
(
K
) for all compact
K
. Arguing as above, for any
A
⊆
R
n
we now have
λ
∗
(
A
+
x
) = sup
{
λ
(
K
)

K
is compact and
K
⊆
A
+
x
}
= sup
{
λ
(
K
)

K
−
x
is compact and
K
−
x
⊆
A
}
= sup
{
λ
(
K
−
x
)

K
−
x
is compact and
K
−
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 Spring '09
 Topology, Continuous function, Metric space, Closed set, E Rn

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