Graphics Reference
In-Depth Information
()
=
()
Ff
xx
if
if
xX
xR X
Œ
n
=
0
Œ
-
.
and choose a rectangle
A
that contains
X
.
Definition.
The
integral
of f over
X
, denoted by Ú
X
f, is defined to be the integral Ú
A
F,
provided that integral is defined. If the integral of f exists, we say that f is
integrable
over
X
.
One can show that whether or not f is integrable on
X
is independent of the rec-
tangle
A
that is chosen.
There is another way to phrase the problem of integrating over an arbitrary set.
Let
X
Õ
R
n
. The function
Definition.
n
c
X
:
RR
Æ
defined by
()
=
c
X
x
1
0
if
if
x X
xR X
Œ
n
(4.20)
=
Œ
-
.
is called the
characteristic function
of
X
.
If the function f above was actually defined over a rectangle
A
, but one simply
wanted to integrate over a smaller set
X
in
A
, then an equivalent definition for the
integral of f over
X
would be to define it to be Ú
A
fc
X
(if that integral exists).
Definition.
A bounded set in
R
n
is said to be
Jordan-measurable
if its boundary is a
set of measure 0.
4.8.2. Theorem.
Let
X
be a Jordan-measurable subset of
R
n
and f :
X
Æ
R
. If f is
bounded on
X
and continuous on
X
except at possibly a set of measure zero, then f
is integrable on
X
.
Proof.
See [Spiv65] or [Buck78].
4.8.3. Corollary.
If
X
is a bounded set in
R
n
, then the characteristic function c
X
of
X
is integrable over
X
if and only if
X
is a Jordan-measurable set.
We can use Theorem 4.8.2 (or its corollary) to define the volume of a set in
R
n
.
Let
X
be a Jordan-measurable set in
R
n
. Define the
volume
of
X
, denoted
Definition.
by vol(
X
), by
()
=
Ú
1.
vol
X
X
X
