We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in \(\mathbb {R}^n\), \(n\ge 2\). To each locally finite Borel measure \(\mu \), we associate a function \(\widetilde{J}_2(\mu ,x)\) which uses a weighted sum to record how closely the mass of \(\mu \) is concentrated near a line in the triples of dyadic cubes containing \(x\). We show that \(\widetilde{J}_2(\mu ,\cdot )<\infty \ \mu \)-a.e. is a necessary condition for \(\mu \) 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.
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