Summary: Let \(E\) be a set in \({\mathbb R}^d\) with finite \(n\)-dimensional Hausdorff measure \({\mathcal H}^n\) such that \(\liminf _{r\to 0}r^{-n}{\mathcal H}^n(B(x,r)\cap E)>0\) for \({\mathcal H}^n\)-a.e. \(x\in E\). In this paper, it is shown that \(E\) is \(n\)-rectifiable if and only if
\[ \int_0^1 \left|\frac{{\mathcal H}^n(B(x,r)\cap E)}{r^n} - \frac{{\mathcal H}^n(B(x,2r)\cap E)}{(2r)^n} \right|^2 \frac{d r}{r}< \infty \; \hbox{ for } \;{\mathcal H}^n\hbox{-a.e. } \;x \in E, \]
and also if and only if
\[ \lim_{r\to0}\left(\frac{{\mathcal H}^n(B(x,r)\cap E)}{r^n} - \frac{{\mathcal H}^n(B(x,2r)\cap E)}{(2r)^n}\right) = 0 \; \hbox{ for } \;{\mathcal H}^n\hbox{-a.e. }\; x \in E. \] Other more general results involving Radon measures are also proved.