Tangent measures of typical measures

TL;DR

This paper demonstrates that for a typical Radon measure in R^d, almost every point has tangent measures that include all non-zero measures, providing a new proof and exploring similar properties for micromeasures on trees.

Contribution

It offers a new self-contained proof of a known result about tangent measures and extends the concept to micromeasures on trees, showing their typical properties.

Findings

Almost every point of a typical Radon measure has all non-zero measures as tangent measures.

The result is sharp; not all measures are tangent measures at some points.

An analogous property is shown for micromeasures on trees.

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.