Rectifiability via a square function and Preiss' theorem

Let $E$ be a set in ${\mathbb{R}}^d$ with finite $n$-dimensional Hausdorff measure $H^n$ such that ${\mathrm{lim\hspace{0.17em}inf}}_{r\to 0}{r}^{-n}{H}^n(B(x,r)\cap E)>0$ for $H^n$-a.e. $x\in E$. In this paper it is shown that $E$ is $n$-rectifiable if and only if $${\int }_0^1{\mid \frac{H^n(B(x,r)\cap E)}{r^n}-\frac{H^n(B(x,2r)\cap E)}{(2r{)}^n}\mid }^2\frac{dr}{r}<\mathrm{\infty }$$ for $H^n$-a.e. $x\in E$; and also if and only if $$\underset{r\to 0}{\lim }(\frac{H^n(B(x,r)\cap E)}{r^n}-\frac{H^n(B(x,2r)\cap E)}{(2r{)}^n})=0$$ for $H^n$-a.e. $x\in E$. Other more general results involving Radon measures are also proved.