Tangent measures of typical measures
arXiv:1203.4221
Abstract
We prove that for a typical Radon measure mu in R^d, every non-zero Radon measure is a tangent measure of mu at mu almost every point. This was already shown by T. O'Neil in his PhD thesis from 1994, but we provide a different self-contained proof for this fact. Moreover, we show that this result is sharp: for any non-zero measure we construct a point in its support where the set of tangent measures does not contain all non-zero measures. We also study a concept similar to tangent measures on trees, micromeasures, and show an analogous typical property for them.
v3: 20 pages, 4 figures, the main result was already proved by T. O'Neil in his PhD thesis with a different proof, added a reference to it. Peer-reviewed version