Consider a Radon measure μ defined on an open subset Ω of n-dimensional Euclidean spaceRn 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
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
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
We denote the set of tangent measures of μ at a by Tan(μ, a).
Existenceedit
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 has positive and finite upper density, i.e. for some .
Propertiesedit
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 .
Examplesedit
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 conceptsedit
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 measureHk 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]
Referencesedit
^Preiss, David (1987). "Geometry of measures in : distribution, rectifiability, and densities". Ann. Math. 125 (3): 537–643. doi:10.2307/1971410. hdl:10338.dmlcz/133417. JSTOR 1971410.
^O'Neil, Toby (1995). "A measure with a large set of tangent measures". Proc. AMS. 123 (7): 2217–2220. doi:10.2307/2160960. JSTOR 2160960.
^Röger, Matthias (2004). "Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation". Interfaces and Free Boundaries. 6 (1): 105–133. doi:10.4171/IFB/93. ISSN 1463-9963. MR 2047075.