s-uniform measures and Marstrand's theorem #
Mattila, Geometry of sets and measures in Euclidean spaces, Chapter 14, uniform measures and Marstrand's theorem.:
- Definition 14.8:
IsSUniform; - Corollary 14.9:
mattila_14_9; - Theorem 14.10 (Marstrand's theorem):
mattila_14_10; - Theorem 14.11:
mattila_14_11.
Definition 14.8 #
Mattila, Definition 14.8. Let s be a positive number. A non-zero Radon measure ν on
ℝⁿ is s-uniform if there is a positive number c such that
0 < ν (B (x, r)) = c r ^ s < ∞ for x ∈ spt ν and 0 < r < ∞.
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Instances For
The set A of Mattila, Corollary 14.9 and Theorem 14.10: the points a at which the
s-dimensional density Θ^s(μ, a) exists and is positive and finite.
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- One or more equations did not get rendered due to their size.
Instances For
Auxiliary results #
A covering comparison: if μ₁ (B (x, r)) ≤ K μ₂ (B (x, r)) for arbitrarily small r at each
point x ∈ S, then μ₁ S ≤ K μ₂ S. (A consequence of the Besicovitch covering theorem.)
The Lebesgue measure of a closed ball in ℝⁿ.
Tangent measures of an s-uniform measure at points of its support are s-uniform and
have 0 in their support.
An s-uniform measure has tangent measures at every point of its support.
There is no s-uniform measure on ℝⁿ with s > n.
An open set G is λ-null for a tangent measure λ ∈ Tan (μ, a) as soon as, at all small
scales r, the blow-up T_{a,r} maps no point of spt μ into G. (This uses the lower
semicontinuity of weak limits on open sets, Theorem 1.24 (2) in Mattila.)
If the support of an s-uniform measure is not all of ℝⁿ, then it has an s-uniform
tangent measure λ with 0 ∈ spt λ ⊆ {x | 0 ≤ x ⬝ e} for some e ≠ 0.
Points carry no mass for an s-uniform measure.
For an s-uniform measure, closed balls of the same radius centred at points of the support
have the same measure (for every real radius).
Mattila, Chapter 14, Exercise 5 (in the generality needed): for an s-uniform measure
ν and x, y ∈ spt ν, ∫_{B (y, r)} g (|z - y|) dν z = ∫_{B (x, r)} g (|z - x|) dν z for every
continuous g. In particular ∫_{B(y,r)} (r² - |z - y|²) dν z = ∫_{B(r)} (r² - |z|²) dν z when
0 ∈ spt ν.
The first case in the proof of Theorem 14.10: if spt ν ⊆ {x | 0 ≤ x ⬝ e} and
∫_{B(r)} x ⬝ e dν x = 0 for all r > 0, then spt ν ⊆ {x | x ⬝ e = 0}.
The last step of the proof of Theorem 14.10: if |y ⬝ m| ≤ C |y|² for all y ∈ spt ν near
0, then every tangent measure λ ∈ Tan (ν, 0) has spt λ ⊆ {y | y ⬝ m = 0}.
The key estimate in the proof of Theorem 14.10: if 0 ∈ spt ν and m = ∫_{B(r)} z dν z,
then |y ⬝ m| ≤ C |y|² for y ∈ spt ν with |y| ≤ r.
The core of the proof of Theorem 14.10: an s-uniform measure ν with
0 ∈ spt ν ⊆ {x | 0 ≤ x ⬝ e} produces an s-uniform measure supported in a hyperplane.
An s-uniform measure on ℝⁿ⁺¹ supported in a hyperplane gives an s-uniform measure on
ℝⁿ.
An s-uniform measure whose support is not all of ℝⁿ⁺¹ gives an s-uniform measure on
ℝⁿ.
If there is an s-uniform measure on ℝⁿ, then s is an integer ≤ n.
A Radon measure on ℝⁿ giving mass c r ^ n to every closed ball of radius r is a
constant multiple of Lebesgue measure.
At points where the density exists and is positive and finite, the density ratio of
Lemma 14.7 is 1, and the point belongs to the set A of Lemma 14.7.
At points of the set A of Lemma 14.7 the measures of small balls are comparable to
ρ ^ s.
Corollary 14.9 #
Mattila, Corollary 14.9 (to Lemma 14.7). Let s be a positive number, μ a Radon
measure on ℝⁿ and A the set of points a ∈ ℝⁿ such that the density Θ^s(μ, a) exists and
is positive and finite. Then for μ almost all a ∈ A every ν ∈ Tan (μ, a) is s-uniform
with 0 ∈ spt ν.
Theorem 14.10 (Marstrand's theorem) #
Mattila, Theorem 14.10 (Marstrand's theorem). Let s be a positive number. Suppose
that there exists a Radon measure μ on ℝⁿ such that the density Θ^s(μ, a) exists and is
positive and finite in a set of positive μ measure. Then s is an integer.
Theorem 14.11 #
Mattila, Theorem 14.11. If ν is an n-uniform measure in ℝⁿ, then ν is a
constant multiple of the Lebesgue measure ℒⁿ.