Tangent measures of typical measures. (English) Zbl 1384.28006

Sahlsten, Tuomas

Real Anal. Exch. 40(2014-2015), No. 1, 53-80 (2015).

Author’s abstract: We prove that for a typical Radon measure \(\mu\) in \(\mathbb{R}^d\), every non-zero Radon measure is a tangent measure of \(\mu\) at \(\mu\) almost every point. This is already shown by T. O’Neil in his Ph.D. thesis form 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.

Reviewer: Ryszard Pawlak (Łódź)

MSC:

28A12

Contents, measures, outer measures, capacities

Keywords:

tangent measures; meagre set; residual set; Radon measure