Consider a Radon measureμ defined on an open subset Ω of n-dimensional Euclidean spaceRn and leta be an arbitrary point in Ω. We can “zoom in” on a small open ball of radius r around a, Br, via the transformation 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 by looking at the push-forward measure defined by where As rgets 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 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 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 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 Rdsuch that, for μ-almost every point a ∈ Rd, Tan consists of all nonzero Radon measures.
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.
