Let $E$ be a set in $\mathbb R^d$ with finite $n$-dimensional Hausdorff measure $H^n$ such that $\liminf_{ r\to0}r^{ -n} H^n(B(x,r)\cap E)>0$ 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 $$ \lim_{ r\to0}\left(\frac{ H^n(B(x,r)\cap E)}{ r^n} - \frac{ H^n(B(x,2r)\cap E)}{ (2r)^n}\right) = 0$$ for $H^n$-a.e. $x\in E$. Other more general results involving Radon measures are also proved.
Categories: math.CA math.AP