Rectifiability via a Square Function and Preiss’ Theorem

Let E be a set in ${\BBR}^d$ with finite n-dimensional Hausdorff measure ${\cal H}^n$ such that $\liminf _{r\to 0}r^{-n}{\cal H}^n(B(x,r)\cap E) \gt 0$ for ${\cal H}^n$-a.e. x∈E. In this paper, it is shown that E is n-rectifiable if and only if $$\int_0^1 \left|{{\cal H}^n(B(x,r)\cap E) \over r^n} - {{\cal H}^n(B(x,2r)\cap E) \over (2r)^n} \right|^2 {{\rm d} r \over r} \lt \infty\quad \hbox{for } {\cal H}^n-\hbox{a.e. } x \in E, $$ and also if and only if $$\lim_{r\to0}\left({{\cal H}^n(B(x,r)\cap E) \over r^n} - {{\cal H}^n(B(x,2r)\cap E) \over (2r)^n}\right) = 0 \quad \hbox{for } {\cal H}^n-\hbox{a.e. } x \in E. $$ Other more general results involving Radon measures are also proved.