Rectifiability via a square function and Preiss' theorem

Abstract

Let $E$ be a set in ${\mathbb{R}}^d$ with finite $n$-dimensional Hausdorff measure $H^n$ such that ${\mathrm{liminf}}_{r\to 0}{r}^{-n}{H}^n(B(x,r)\cap E)>0$ for $H^n$-a.e. $x\in E$. In this paper it is shown that $E$ is $n$-rectifiable if and only if $${\int }_0^1{\left|\frac{H^n(B(x,r)\cap E)}{r^n}-\frac{H^n(B(x,2r)\cap E)}{(2r{)}^n}\right|}^2\frac{dr}{r}<\infty$$ for $H^n$-a.e. $x\in E$; and also if and only if $$\underset{r\to 0}{\lim }\left(\frac{H^n(B(x,r)\cap E)}{r^n}-\frac{H^n(B(x,2r)\cap E)}{(2r{)}^n}\right)=0$$ for $H^n$-a.e. $x\in E$. Other more general results involving Radon measures are also proved.

Minor corrections and adjustments