Tangent measure
In
Definition
Consider a Radon measure μ defined on an
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
where
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
- where the limit is taken in the compact supportin 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 doublingat a, i.e.
- μ has positive and finite upper density, i.e. for some .
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 and , then .
- 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 and , then .
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 and x is in the support of ν, then .
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 Rd such that, for μ-almost every point a ∈ Rd, Tan(μ, a) consists of all nonzero Radon measures.[2]
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 Hk on P. More precisely:
- Definition. P is the k-dimensional tangent space of μ at a if there is a θ > 0 such that
- where μa,r is the translated and rescaled measure given by
- The number θ is called the multiplicity of μ at a, and the tangent space of μ at a is denoted Ta(μ).
Further study of tangent measures and tangent spaces leads to the notion of a varifold.[3]