Tangent measure

Definition

In measure theory, tangent measures are used to study the local behavior of Radon measures, analogous to how tangent spaces are used to study the local behavior of differentiable manifolds.

We start by considering a Radon measure $\mu$ defined on the n-dimensional Euclidean space $\mathbb {R} ^{n}$ and consider an arbitrary point in the space a. We can zoom up on a small ball of radius r around a, $B_{r}(a)$, via the transformation

$T_{a,r}(x)={\frac {x-a}{r}}$

which enlarges the ball of radius r about a to a ball of radius 1 centered at 0. With this, we may now ‘’zoom’’ up on how $\mu$ behaves on $B_{r}(a)$ by looking at the measure defined as

$T_{a,r\#}\mu (A)=\mu (a+rA)$

where $a+rA=\{a+rx:x\in A\}$. As ‘’r’’ gets smaller, this transformation on the measure $\mu$ spreads out and enlarges the portion of $\mu$ 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 $\mu$ at the point a is a Radon measure $\nu$ such that there exist sequences of positive numbers $c_{i}>0$ and decreasing radii $r_{i}\rightarrow 0$ such that

$\lim _{i\rightarrow \infty }c_{i}T_{a,r_{i}\#}\mu =\nu$

where the limit is taken in the weak-star topology, i.e., for any continuous function $\varphi$ with compact support in $\mathbb {R} ^{n}$,

$\lim _{i\rightarrow \infty }\int \varphi d(c_{i}T_{a,r_{i}\#}\mu )=\int \varphi d\nu .$

We denote the set of tangent measures of $\mu$ at a by $T(\mu ,a)$.

Examples

• Suppose we have a circle in $\mathbb {R} ^{2}$. 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.

Discussion

• Tangent measures are a useful tool in geometric measure theory. For example, they are used in proving Martstrand’s theorem.