Tangent measure

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, B_r(a), via the transformation

T a , r ( x ) = 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 B_r(a) by looking at the push-forward measure defined by

T a , r # μ ( A ) = μ ( a + r A )

where

a + r A = { a + r x : x ∈ 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 c_i > 0 and decreasing radii r_i → 0 such thatwhere the limit is taken in the weak-∗ topology, i.e., for any continuous function φ with compact support in Ω,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 ) ) < ∞

μ has positive and finite upper density, i.e. 0 < lim sup r ↓ 0 μ ( B ( a , r ) ) r s < ∞ for some 0 < s < ∞ .

Properties

The collection of tangent measures at a point is closed under two types of scaling. Cones of measures were also defined by Preiss.

The set Tan(μ, a) of tangent measures of a measure μ at a point a in the support of μ is a cone of measures, i.e. if ν ∈ T a n ( μ , a ) and c > 0 , then c ν ∈ T a n ( μ , a ) .

The cone Tan(μ, a) of tangent measures of a measure μ at a point a in the support of μ is a d-cone or dilation invariant, i.e. if ν ∈ T a n ( μ , a ) and r > 0 , then T 0 , r # ν ∈ T a n ( μ , a ) .

At typical points in the support of a measure, the cone of tangent measures is also closed under translations.

At μ almost every a in the support of μ, the cone Tan(μ, a) of tangent measures of μ at a is translation invariant, i.e. if ν ∈ T a n ( μ , a ) and x is in the support of ν, then T x , 1 # ν ∈ T a n ( μ , a ) .

Examples

Suppose we have a circle in R² with uniform measure on that circle. Then, for any point a in the circle, the set of tangent measures will just be positive constants times 1-dimensional Hausdorff measure supported on the line tangent to the circle at that point.

In 1995, Toby O'Neil produced an example of a Radon measure μ on R^d such that, for μ-almost every point a ∈ R^d, Tan(μ, a) consists of all nonzero Radon measures.

Related concepts

There is an associated notion of the tangent space of a measure. A k-dimensional subspace P of Rⁿ is called the k-dimensional tangent space of μ at a ∈ Ω if — after appropriate rescaling — μ “looks like” k-dimensional Hausdorff measure H^k on P. More precisely:

Definition. P is the k-dimensional tangent space of μ at a if there is a θ > 0 such thatwhere μ_a,r is the translated and rescaled measure given byThe number θ is called the multiplicity of μ at a, and the tangent space of μ at a is denoted T_a(μ).

Further study of tangent measures and tangent spaces leads to the notion of a varifold.