Multiscale analysis of 1-rectifiable measures: necessary conditions

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in (Formula presented.), (Formula presented.). To each locally finite Borel measure (Formula presented.), we associate a function (Formula presented.) which uses a weighted sum to record how closely the mass of (Formula presented.) is concentrated near a line in the triples of dyadic cubes containing (Formula presented.). We show that (Formula presented.)-a.e. is a necessary condition for (Formula presented.) 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.