In measure theory, a field of mathematics, the Hausdorff density measures how concentrated a Radon measure is at some point.
Let
μ
be a Radon measure and
a
∈
R
n
some point in Euclidean space. The s-dimensional upper and lower Hausdorff densities are defined to be, respectively,
Θ
∗
s
(
μ
,
a
)
=
lim sup
r
→
0
μ
(
B
r
(
a
)
)
r
s
and
Θ
∗
s
(
μ
,
a
)
=
lim inf
r
→
0
μ
(
B
r
(
a
)
)
r
s
where
B
r
(
a
)
is the ball of radius r > 0 centered at a. Clearly,
Θ
∗
s
(
μ
,
a
)
≤
Θ
∗
s
(
μ
,
a
)
for all
a
∈
R
n
. In the event that the two are equal, we call their common value the s-density of
μ
at a and denote it
Θ
s
(
μ
,
a
)
.
Marstrand's theorem
The following theorem states that the times when the s-density exists are rather seldom.
Marstrand's theorem: Let
μ
be a Radon measure on
R
d
. Suppose that the
s-density
Θ
s
(
μ
,
a
)
exists and is positive and finite for
a in a set of positive
μ
measure. Then
s is an integer.
In 1987 David Preiss proved a stronger version of Marstrand's theorem. One consequence is that sets with positive and finite density are rectifiable sets.
Preiss' theorem: Let
μ
be a Radon measure on
R
d
. Suppose that
m
≥
1
is an integer and the
m-density
Θ
m
(
μ
,
a
)
exists and is positive and finite for
μ
almost every
a in the support of
μ
. Then
μ
is
m-rectifiable, i.e.
μ
≪
H
m
(
μ
is absolutely continuous with respect to Hausdorff measure
H
m
) and the support of
μ
is an
m-rectifiable set.
