A doubling measure on $\R^d$ can charge a rectifiable curve

TL;DR

The paper demonstrates that in dimensions two and higher, there exists a doubling measure that assigns positive measure to a rectifiable curve, challenging previous assumptions about measure and rectifiability.

Contribution

It constructs a specific doubling measure on  that charges a rectifiable curve, showing a new interaction between measure doubling properties and rectifiability.

Findings

Existence of a doubling measure charging a rectifiable curve in 

Construction method for such a measure and curve

Implications for measure theory and geometric analysis

Abstract

For $d\geq 2$, we construct a doubling measure $\nu$ on $\R^d$ and a rectifiable curve $\Gamma$ such that $\nu(\Gamma)>0$.