REVIEW OF LEBESGUE MEASURE AND INTEGRATION
CHRISTOPHER HEIL
These notes will briefly review some basic concepts related to the theory of Lebesgue
measure and the Lebesgue integral.
We are not trying to give a complete development,
but rather review the basic definitions and theorems with at most a sketch of the proof of
some theorems. These notes follow the text
Measure and Integral
by R. L. Wheeden and
A. Zygmund, Dekker, 1977, and full details and proofs can be found there.
1.
OPEN, CLOSED, AND COMPACT SUBSETS OF EUCLIDEAN SPACE
Notation 1.1.
N
=
{
1
,
2
,
3
, . . .
}
is the set of natural numbers,
Z
=
{
. . . ,
−
1
,
0
,
1
, . . .
}
is the
set of integers,
Q
is the set of rational numbers,
R
is the set of real numbers, and
C
is the
set of complex numbers.
R
d
is real
d
dimensional Euclidean space, the space of all vectors
x
= (
x
1
, . . . , x
d
) with
x
1
, . . . , x
d
∈
R
.
On occasion, we formally use the
extended real number line
R
∪ {−∞
,
∞}
= [
−∞
,
∞
],
but it is important to note that
∞
is a formal object, not a number. To write
a
∈
[
−∞
,
∞
]
means that either
a
is a finite real number or
a
is one of
±∞
. We write

a

<
∞
to mean
that
a
is a finite real number. Note that there is no analogue of the extended reals when we
consider complex numbers; there’s no obvious “
∞
” or “
−∞
.”
We declare some arithmetic conventions for the extended real numbers:
∞
+
∞
=
∞
,
1
/
0 =
∞
, 1
/
∞
= 0, and 0
· ∞
= 0. The symbols
∞ − ∞
are undefined, i.e., they have no
meaning.
The empty set is denoted by
∅
. Two sets
A
,
B
are
disjoint
if
A
∩
B
=
∅
. A collection
{
A
k
}
of sets are disjoint if
A
j
∩
A
k
=
∅
whenever
j
negationslash
=
k
.
The
real part
of a complex number
z
=
a
+
ib
is Re (
z
) =
a
, and the
imaginary part
is
Im (
z
) =
b
. The
complex conjugate
of
z
=
a
+
ib
is ¯
z
=
a
−
ib
. The
modulus
, or
absolute
value
, of
z
=
a
+
ib
is

z

=
√
z
¯
z
=
√
a
2
+
b
2
.
square
For concreteness, we will use the Euclidean distance on
R
d
in these notes. However, all
the results of this section are valid with respect to any norm on
R
d
.
Definition 1.2.
(a) The
Euclidean norm
on
R
d
is

x

= (
x
2
1
+
· · ·
+
x
2
d
)
1
/
2
.
The
distance
between
x
,
y
∈
R
d
is

x
−
y

.
Date
: Revised July 30, 2007.
c
circlecopyrt
2007 by Christopher Heil.
1
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REVIEW OF LEBESGUE MEASURE AND INTEGRATION
(b) Suppose that
{
x
n
}
n
∈
N
is a sequence of points in
R
d
and that
x
∈
R
d
. We say that
x
n
converges
to
x
, and write
x
n
→
x
or
x
= lim
n
→∞
x
n
, if
lim
n
→∞

x
n
−
x

= 0
.
square
Using this definition of distance, we can now define open and closed sets in
R
d
and state
some of their basic properties.
Definition 1.3.
Let
E
⊆
R
d
be given.
(a)
E
is
open
if for each point
x
∈
E
there is some open ball
B
r
(
x
) =
{
y
∈
R
d
:

x
−
y

< r
}
centered at
x
that is completely contained in
E
, i.e.,
B
r
(
x
)
⊆
E
for some
r >
0.
(b) A point
x
∈
R
d
is a
limit point
of
E
if there exist points
x
n
∈
E
that converge to
x
,
i.e., such that
x
n
→
x
.
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