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 (introduced by David Preiss in his study of rectifiable sets) are a useful tool in geometric measure theory. For example, they are used in proving Marstrand’s theorem and Preiss' theorem. 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 (introduced by David Preiss in his study of rectifiable sets) are a useful tool in geometric measure theory. For example, they are used in proving Marstrand’s theorem and Preiss' theorem. 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, 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