{"paper":{"title":"The descriptive set theory of the Lebesgue density theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Alessandro Andretta, Riccardo Camerlo","submitted_at":"2011-05-17T12:39:34Z","abstract_excerpt":"Given an equivalence class $[A]$ in the measure algebra of the Cantor space, let $\\hat\\Phi([A])$ be the set of points having density 1 in $A$. Sets of the form $\\hat\\Phi([A])$ are called $\\mathcal{T}$-regular. We establish several results about $\\mathcal{T}$-regular sets. Among these, we show that $\\mathcal{T}$-regular sets can have any complexity within $\\Pi^{0}_{3}$ (=$ \\mathbf{F}_{\\sigma\\delta}$), that is for any $\\Pi^{0}_{3}$ subset $X$ of the Cantor space there is a $\\mathcal{T}$-regular set that has the same topological complexity of $X$. Nevertheless, the generic $\\mathcal{T}$-regular s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.3355","kind":"arxiv","version":1},"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"}