In measure theory, tangent measures are used to study the local behavior of Radon measures, in much the same way as tangent spaces are used to study the local behavior of differentiable manifolds. Tangent measures (introduced by David Preiss in his study of rectifiable sets) are a useful tool in geometric measure theory. For example, they are used in proving Marstrand's theorem and Preiss' theorem.
Definition Consider a Radon measure μ defined on an open subset Ω of n-dimensional Euclidean space Rn and let a be an arbitrary point in Ω. We can "zoom in" on a small open ball of radius r around a, Br(a), via the transformation
T a , r ( x ) = x − a r , {\displaystyle T_{a,r}(x)={\frac {x-a}{r}},}
which enlarges the ball of radius r about a to a ball of radius 1 centered at 0. With this, we may now zoom in on how μ behaves on Br(a) by looking at the push-forward measure defined by
T a , r # μ ( A ) = μ ( a + r A ) {\displaystyle T_{a,r\#}\mu (A)=\mu (a+rA)}
where
a + r A = { a + r x : x ∈ A } . {\displaystyle a+rA=\{a+rx:x\in A\}.}
As r gets smaller, this transformation on the measure μ spreads out and enlarges the portion of μ supported around the point a. We can get information about our measure around a by looking at what these measures tend to look like in the limit as r approaches zero.
Definition. A tangent measure of a Radon measure μ at the point a is a second Radon measure ν such that there exist sequences of positive numbers ci > 0 and decreasing radii ri → 0 such that
lim i → ∞ c i T a , r i # μ = ν {\displaystyle \lim _{i\rightarrow \infty }c_{i}T_{a,r_{i}\#}\mu =\nu }
where the limit is taken in the weak-∗ topology, i.e., for any continuous function φ with compact support in Ω,
lim i → ∞ ∫ Ω φ d ( c i T a , r i # μ ) = ∫ Ω φ d ν . {\displaystyle \lim _{i\rightarrow \infty }\int _{\Omega }\varphi \,\mathrm {d} (c_{i}T_{a,r_{i}\#}\mu )=\int _{\Omega }\varphi \,\mathrm {d} \nu .}
We denote the set of tangent measures of μ at a by Tan(μ, a).
Existence The set Tan(μ, a) of tangent measures of a measure μ at a point a in the support of μ is nonempty on mild conditions on μ. By the weak compactness of Radon measures, Tan(μ, a) is nonempty if one of the following conditions hold:
μ is asymptotically doubling at a, i.e. lim sup r ↓ 0 μ ( B ( a , 2 r ) ) μ ( B ( a , r ) ) < ∞ {\displaystyle \limsup _{r\downarrow 0}{\frac {\mu (B(a,2r))}{\mu (B(a,r))}}<\infty }
μ has positive and finite upper density, i.e. 0 < lim sup r ↓ 0 μ ( B ( a , r ) ) r s < ∞ {\displaystyle 0<\limsup _{r\downarrow 0}{\frac {\mu (B(a,r))}{r^{s}}}<\infty } for some 0 < s < ∞ {\displaystyle 0<s<\infty } .
Properties The collection of tangent measures at a point is closed under two types of scaling. Cones of measures were also defined by Preiss.
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