Stratification of the single blow-up set for Radon measures

TL;DR

This paper proves that the set of points with unique blow-up and a specific invariant subspace dimension for Radon measures is rectifiable, providing new criteria and extending existing results on measure rectifiability.

Contribution

It establishes the rectifiability of points with unique blow-up in Radon measures and extends prior results on measure behavior almost everywhere.

Findings

The set of points with unique blow-up and fixed invariant subspace dimension is k-rectifiable.

Provides a new rectifiability criterion for signed Radon measures.

Extends Mattila's result on measures with unique blow-up almost everywhere.

Abstract

We show that the set of points where the blow-up, in the sense of Preiss, of a signed Radon measure on $\mathbb{R}^n$ is unique and its invariant subspace has dimension $k$ is $k$-rectifiable. As simple applications, we obtain a rectifiability criterion for signed Radon measures and the extension of a result, due to Mattila, on measures having unique blow-up almost everywhere.