Tangent measure
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 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, via the transformationwhich 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 by looking at the push-forward measure defined by
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.
Existence
The set Tan 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 is nonempty if one of the following conditions hold:μ is asymptotically doubling at 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 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 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 μ almost every a in the support of μ, the cone Tan 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 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 Hk on P. More precisely:Further study of tangent measures and tangent spaces leads to the notion of a varifold.