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

We show that a Radon measure μ in 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 μ -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, and we show an application to the study of analytic capacity.