{"paper":{"title":"A characterization of $1$-rectifiable doubling measures with connected supports","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.MG","authors_text":"Jonas Azzam, Mihalis Mourgoglou","submitted_at":"2015-01-09T18:33:11Z","abstract_excerpt":"Garnett, Killip, and Schul have exhibited a doubling measure $\\mu$ with support equal to $\\mathbb{R}^{d}$ which is $1$-rectifiable, meaning there are countably many curves $\\Gamma_{i}$ of finite length for which $\\mu(\\mathbb{R}^{d}\\backslash \\bigcup \\Gamma_{i})=0$. In this note, we characterize when a doubling measure $\\mu$ with support equal to a connected metric space $X$ has a $1$-rectifiable subset of positive measure and show this set coincides up to a set of $\\mu$-measure zero with the set of $x\\in X$ for which $\\liminf_{r\\rightarrow 0} \\mu(B_{X}(x,r))/r>0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.02220","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}