A measure is 1-recti able if there is a countable union of nite length curves whose complement has zero measure. We characterize 1-recti able Radon measures µ in n-dimensional Euclidean space for all n ≥ in terms of positivity of the lower density and niteness of a geometric square function, which loosely speaking, records in an L gauge the extent to which µ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between µ and 1-dimensional Hausdor measure H . We also characterize purely 1-unrecti able Radon measures, i.e. locally nite measures that give measure zero to every nite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L variant of P. Jones' traveling salesman construction, which is of independent interest.

Abstract. We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in R n , n ≥ 2. To each locally finite Borel measure µ, we associate a function J 2 (µ, x) which uses a weighted sum to record how closely the mass of µ is concentrated near a line in the triples of dyadic cubes containing x. We show that J 2 (µ, ·) < ∞ µ-a.e. is a necessary condition for µ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure.Thus we are able to analyze general 1-rectifiable measures, including measures which are singular with respect to 1-dimensional Hausdorff measure.

We show that on an NTA domain if each tangent measure to harmonic measure at a point is a polynomial harmonic measure then the associated polynomials are homogeneous. Geometric information for solutions of a two-phase free boundary problem studied by Kenig and Toro is derived.

We show the David-Jerison construction of big pieces of Lipschitz graphs inside a corkscrew domain does not require surface measure be upper Ahlfors regular. Thus we can study absolute continuity of harmonic measure and surface measure on NTA domains of locally finite perimeter using Lipschitz approximations. A partial analogue of the F. and M. Riesz Theorem for simply connected planar domains is obtained for NTA domains in space. As one consequence every Wolff snowflake has infinite surface measure.