{"paper":{"title":"Measuring sets with translation invariant Borel measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.FA","authors_text":"Andr\\'as M\\'ath\\'e","submitted_at":"2015-04-10T19:24:26Z","abstract_excerpt":"Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets $B$ in $\\mathbb{R}$ (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and \\sigma-finite measure to $B$. We investigate which sets can be written as a (disjoint) union of measured sets.\n  We show that every Borel nullset $B\\subset \\mathbb{R}$ of the second category is larger than any nullset $A\\subset \\mathbb{R}$ in the sense that there are partitions $B=B_1\\cup B_2$, $A=A_1\\cup A_2$ and gauge functions $g_1, g_2$ such that the Hausdorff measures satisfy $H^{g_i}(B_i)=1$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.02765","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"}