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A measure is 1-rectifiable if there is a countable union of finite lengthcurves whose complement has zero measure. We characterize 1-rectifiable Radonmeasures $\mu$ in $n$-dimensional Euclidean space for all $n\geq 2$ in terms ofpositivity of the lower density and finiteness of a geometric square function,which loosely speaking, records in an $L^2$ gauge the extent to which $\mu$admits approximate tangent lines, or has rapidly growing density ratios, alongits support. In contrast with the classical theorems of Besicovitch, Morse andRandolph, and Moore, we do not assume an a priori relationship between $\mu$and 1-dimensional Hausdorff measure. We also characterize purely1-unrectifiable Radon measures, i.e. locally finite measures that give measurezero to every finite length curve. Characterizations of this form wereoriginally conjectured to exist by P. Jones. Along the way, we develop an $L^2$variant of P. Jones' traveling salesman construction, which is of independentinterest.

Multiscale analysis of 1-rectifiable measures II: characterizations