{"paper":{"title":"Haar null sets and the consistent reflection of non-meagreness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.LO"],"primary_cat":"math.CA","authors_text":"Juris Stepr\\=ans, M\\'arton Elekes","submitted_at":"2011-09-28T10:55:23Z","abstract_excerpt":"A subset $X$ of a Polish group $G$ is called \\emph{Haar null} if there exists a Borel set $B \\supset X$ and Borel probability measure $\\mu$ on $G$ such that $\\mu(gBh)=0$ for every $g,h \\in G$. We prove that there exists a set $X \\subset \\mathbb{R}$ that is not Lebesgue null and a Borel probability measure $\\mu$ such that $\\mu(X + t) = 0$ for every $t \\in \\mathbb{R}$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.)\n  "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.6164","kind":"arxiv","version":2},"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"}