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provided that this limit exists.
We should point out that we are using Barnsley's terminology ([Barn88]) here, but
there does not seem to be any universal agreement for the name “fractal dimension.”
For example, in Alligood et al. ([AlSY97]) it is called the “box-counting dimension.”
Alligood et al. use d-dimensional boxes instead of closed balls, but that is an unim-
portant difference.
We shall describe one more notion of dimension. Although we restrict ourselves
to subsets of
R
n
, what we say next applies to subsets of an arbitrary metric space.
Let
X
be a nonempty bounded subset of
R
n
. Let p be an arbitrary real number, 0
£ p <•. If 0 <e, define
•
Â
•
Ï
Ó
¸
˛
p
I
(
)
=
(
()
)
()
<
m
X
,
e
inf
diam
U
U
Ã
X
,
U
=
X
,
and diam
U
e
p
i
i
i
i
i
=
1
i
=
1
and
()
=
{
(
)
}
m
X
sup
m
X
,
ee0
>
.
p
p
Definition.
The quantity m
p
(
X
) is called the
Hausdorff p-dimensional measure
of
X
.
The interesting property of the function m
p
(
X
) is that, as a function of p, it
assumes only three possible values, namely, 0, •, or a
single
finite value.
22.3.2
Theorem.
There is a unique real number d
H
, 0 £ d
H
£ n, so that
()
=•
m
if p
<
d
p
H
=
0
if p
>
d
.
H
Proof.
See [Fede69] or [Falc85].
Definition.
The number d
H
appearing in Theorem 22.3.2 is called the
Hausdorff-
Besicovitch dimension
of
X
and will be denoted by dim
H
X
.
The various dimensions can be related.
If
X
is a bounded subset of
R
n
, then dim
X
£ dim
H
X
£ dim
F
X
£ n.
22.3.3
Theorem.
Proof.
See [HurW48] and [Barn88].
On “nice” spaces these definitions of dimension lead to the same
integer
dimen-
sion. One final definition of dimension will be given in the next section.
What we have here is a good example of where some topic that people used to
think was of interest only to hardcore mathematicians, all of a sudden got some very
practical importance. After all, who would have thought that weird spaces with non-
integer dimensions would ever be relevant to the “real” world. Of course, most people
might not even have been aware of weird-dimensional spaces because dimension
theory is typically only encountered by graduate students in mathematics. Perhaps a
