# Multiscale Analysis of 1-rectifiable Measures II: Characterizations

@article{Badger2016MultiscaleAO, title={Multiscale Analysis of 1-rectifiable Measures II: Characterizations}, author={Matthew Badger and Raanan Schul}, journal={Analysis and Geometry in Metric Spaces}, year={2016}, volume={5}, pages={1 - 39} }

Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the…

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