  In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable
curves in the plane via a multiscale sum of $\beta$-numbers. These
$\beta$-numbers are geometric quantities measuring how far a given set deviates
from a best fitting line at each scale and location. Jones' result is a
quantitative way of saying that a curve is rectifiable if and only if it has a
tangent at almost every point. Moreover, computing this square sum for a curve
returns the length of the curve up to multiplicative constant. K. Okikiolu
extended his result from subsets of the plane to subsets of Euclidean space. G.
David and S. Semmes extended the discussion to include sets of (integer)
dimension larger than one, under the assumption of Ahlfors regularity and using
a variant of Jones' $\beta$ numbers. In this paper we give a version of P.
Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean
space. We estimate the $d$-dimensional Hausdorff measure of a set in terms of
an analogous sum of $\beta$-type numbers. There is no assumption of Ahlfors
regularity, but rather, only of a lower bound on the Hausdorff content. We
adapt David and Semmes' version of Jones' $\beta$-numbers by redefining them
using a Choquet integral. A key tool in the proof is G. David and T. Toro's
parametrization of Reifenberg flat sets (with holes).
