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.

Definition

Consider a Radon measure "μ" defined on an open subset Ω of "n"-dimensional Euclidean space R^"n" 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: $T\_\{a,r\}(x)=frac\{x-a\}\{r\},$which enlarges the ball of radius "r" about "a" to a ball of radius 1 centered at 0. With this, we may now zoom in on how "μ" behaves on "B"_"r"("a") by looking at the push-forward measure defined by:$T\_\{a,r\; \#\}mu(A)=mu(a+rA)$where:$a+rA=\{a+rx:xin\; A\}.$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

:: $lim\_\{i\; ightarrowinfty\}\; c\_\{i\}T\_\{a,r\_\{i\}\#\}mu\; =\; u$

: where the limit is taken in the weak-∗ topology, i.e., for any continuous function "φ" with compact support in Ω,

:: $lim\_\{i\; ightarrowinfty\}int\_\{Omega\}\; varphi\; ,\; mathrm\{d\}\; (c\_\{i\}T\_\{a,r\_\{i\}\#\}mu)=int\_\{Omega\}\; varphi\; ,\; mathrm\{d\}\; u.$

:We denote the set of tangent measures of "μ" at "a" by T("μ", "a").

There is an associated notion of the tangent space of a measure. A "k"-dimensional subspace "P" of R^"n" is called the "k"-dimensional tangent space of "μ" at "a" ∈ Ω if — after appropriate rescaling — "μ" “looks like” "k"-dimensional Hausdorff measure "H"^"k" on "P". More precisely:

:Definition. "P" is the "k"-"dimensional tangent space" of "μ" at "a" if there is a "θ" > 0 such that

::$mu\_\{a,\; r\}\; xrightarrow\; [r\; o\; 0]\; \{*\}\; heta\; H^\{k\}\; lfloor\_\{P\},$

:where "μ"_"a","r" is the translated and rescaled measure given by

::$mu\_\{a,\; r\}\; (A)\; =\; frac1\{r^\{n\; -\; 1\; mu(a\; +\; r\; A).$

:The number "θ" is called the "multiplicity" of "μ" at "a", and the tangent space of "μ" at "a" is denoted T_"a"("μ").

Further study of tangent measures and tangent spaces leads to the notion of a varifold. [cite journal | last = Röger | first = Matthias | title = Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation | journal = Interfaces Free Bound. | volume = 6 | year = 2004 | issue = 1 | pages = 105–133 | issn = 1463-9963 MathSciNet|id=2047075]

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 R^"d" such that, for almost every point "a" ∈ R^"d", T("μ", "a") consists of all nonzero Radon measures. [cite journal
last = O'Neil | first = Toby
title = A measure with a large set of tangent measures
journal = Proc. of the AMS | volume = 123 | pages = 2217–2220 | year = 1995
doi = 10.2307/2160960]

References