Tangent Measure - Definition

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

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

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 c_i > 0 and decreasing radii r_i → 0 such that

where 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).