1
Measure
1.1
Rectangles
Almost Disjoint.
A union of rectangles is said to be
almost disjoint
if the interiors of the rectangles
are disjoint.
Lemma 1.1.
If a rectangle is the almost disjoint union of finitely many other rectangles, say
R
=
S
N
k
=1
R
k
, then

R

=
∑
N
k
=1

R
k

.
Lemma 1.2.
If
R, R
1
, . . . , R
N
are rectangles and
R
⊂
S
N
k
=1
R
k
, then

R
 ≤
∑
N
k
=1

R
k

.
Theorem 1.3.
Every open subset
O
of
can be written uniquely as a countable union of disjoint open
intervals.
Theorem 1.4.
Every open subset
O
of
d
,
d
≥
1, can be written as a countable union of almost
disjoint closed cubes.
1.2
Outer Measure
Exterior Measure.
For any subset
E
∈
d
, the
exterior measure
of
E
is
m
*
(
E
) = inf
∑
∞
j
=1

Q
j

,
where the infimum is taken over all countable coverings
E
⊂
S
∞
i
=1
Q
j
.
Observation 1 (Monotonicity).
If
E
1
⊂
E
2
, then
m
*
(
E
1
)
≤
m
*
(
E
2
).
Observation 2 (Countable subadditivity).
If
E
=
S
∞
i
=1
E
j
, then
m
*
(
E
)
≤
∑
∞
i
=1
m
*
E
j
.
Observation 3.
If
E
⊂
d
, then
m
*
(
E
) = inf
m
*
(
O
), where the infimum is taken over all open sets
O
containing
E
.
Observation 4.
If
E
=
E
1
∪
E
2
, and
d
(
E
1
, E
2
)
>
0, then
m
*
(
E
) =
m
*
(
E
1
) +
m
*
(
E
2
).
Observation 5.
If a set
E
is the countable union of almost disjoint cubes
E
=
S
∞
i
=1
Q
j
, then
m
*
(
E
) =
∞
X
i
=1

Q
j

.
1.3
Lebesgue Measure
Measurable Set.
A subset
E
is
Lebesgue measurable
if for any
>
0 there exists an open set
O
with
E
⊂ O
and
m
*
(
O 
E
)
<
.
If
E
is measurable, we define its
Lebesgue measure
to be
m
(
E
) =
m
*
(
E
).
Property 1.
Every open set in
d
is measurable.
Property 2.
If
m
*
(
E
) = 0, then
E
is measurable. In particular, if
F
is a subset of a set of exterior
measure 0, then
F
is measurable.
Property 3.
A countable union of measurable sets is measurable.
Property 4.
Closed sets are measurable.
Property 5.
The complement of a measurable set is measurable.
Property 6.
A countable intersection of measurable sets is measurable.
Lemma 3.1.
If
F
is closed,
K
is compact, and these sets are disjoint, then
d
(
F, K
)
>
0.
Theorem 3.2.
If
E
1
, E
2
, . . .
, are disjoint measurable sets, and
E
=
S
∞
i
=1
E
j
, then
m
(
E
) =
∑
∞
i
=1
m
(
E
i
).
Corollary 3.3.
Suppose
E
1
, E
2
, . . .
are measurable subsets of
d
.
1. If
E
k
%
E
, then
m
(
E
) = lim
N
→∞
m
(
E
N
).
2. If
E
k
&
E
and
m
(
E
k
)
<
∞
for some
k
, then
m
(
E
) = lim
N
→∞
m
(
E
n
).
Theorem 3.4.
Suppose
E
is a measurable subset of
d
. Then, for every
>
0:
1. There exists an open set
O
with
E
⊂ O
and
m
(
O 
E
)
≤
.
2. There exists a closed set
F
with
F
⊂
E
and
m
(
E

F
)
≤
.
3. If
m
(
E
) is finite, there exists a compact set
K
with
K
⊂
E
and
m
(
E

K
)
≤
.
4. If
m
(
E
) is finite, there exists a finite union
F
=
S
N
j
=1
Q
j
of closed cubes such that
m
(
E
4
F
)
≤
.
Corollary 3.5.
A subset
E
of
d
is measurable
1. if and only if
E
=
V
\
N
where
V
is a
G
δ
set and
m
(
N
) = 0,
2. if and only if
E
=
H
∪
M
where
H
is an
F
σ
set and
m
(
M
) = 0.
1