# Multiscale Analysis of 1-rectifiable Measures II: Characterizations

@article{Badger2016MultiscaleAO,
title={Multiscale Analysis of 1-rectifiable Measures II: Characterizations},
journal={Analysis and Geometry in Metric Spaces},
year={2016},
volume={5},
pages={1 - 39}
}
• Published 11 February 2016
• Mathematics
• Analysis and Geometry in Metric Spaces
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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