Characterization of $n$-rectifiability in terms of Jones' square function: Part II

Abstract: We show that a Radon measure $\mu$ in $\mathbb R^d$ which is absolutely continuous with respect to the $n$-dimensional Hausdorff measure $H^n$ is $n$-rectifiable if the so called Jones' square function is finite $\mu$-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all $n$-rectifiable measures which are absolutely continuous with respect to $H^{n}$. Further, in this paper we also investigate the relationship between the Jones' square function and the so called Menger curvature of a measure with linear growth.