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We call a set $X$ naively Haar null if there exists a Borel probability measure $\\mu$ such that for every $g, h \\in G$ we have $\\mu(gXh)=0$.\n  Generalizing a result of Elekes and Stepr\\=ans, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1508.02227","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-08-10T12:57:53Z","cross_cats_sorted":[],"title_canon_sha256":"6ab8ac1304bd717bd6b2004cd199265f7cadf863c35f9332921392c5b9552f11","abstract_canon_sha256":"16eb9d504415e388a2e77c35568d401584b94428e8f16087f1e135c4d446dba8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:33.821159Z","signature_b64":"raBMATw9vnDe/1GSPuCqR0Rm5JNacyGDNFdK9P5WRE6Mytr9lX3qV2VQ+jpfzHe8ZxmZxIaBdW7jJhM0PLZ0BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eaf3b62a0d6f270ab6c20cc7d44af5b9c3039fddf97e1a48bce6ab9583ed537f","last_reissued_at":"2026-05-18T01:35:33.820570Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:33.820570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Naively Haar null sets in Polish groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"M\\'arton Elekes, Zolt\\'an Vidny\\'anszky","submitted_at":"2015-08-10T12:57:53Z","abstract_excerpt":"Let $(G,\\cdot)$ be a Polish group. We say that a set $X \\subset G$ is Haar null if there exists a universally measurable set $U \\supset X$ and a Borel probability measure $\\mu$ such that for every $g, h \\in G$ we have $\\mu(gUh)=0$. We call a set $X$ naively Haar null if there exists a Borel probability measure $\\mu$ such that for every $g, h \\in G$ we have $\\mu(gXh)=0$.\n  Generalizing a result of Elekes and Stepr\\=ans, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02227","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1508.02227","created_at":"2026-05-18T01:35:33.820650+00:00"},{"alias_kind":"arxiv_version","alias_value":"1508.02227v1","created_at":"2026-05-18T01:35:33.820650+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.02227","created_at":"2026-05-18T01:35:33.820650+00:00"},{"alias_kind":"pith_short_12","alias_value":"5LZ3MKQNN4TQ","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_16","alias_value":"5LZ3MKQNN4TQVNWC","created_at":"2026-05-18T12:29:05.191682+00:00"},{"alias_kind":"pith_short_8","alias_value":"5LZ3MKQN","created_at":"2026-05-18T12:29:05.191682+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5LZ3MKQNN4TQVNWCBTD5ISXVXH","json":"https://pith.science/pith/5LZ3MKQNN4TQVNWCBTD5ISXVXH.json","graph_json":"https://pith.science/api/pith-number/5LZ3MKQNN4TQVNWCBTD5ISXVXH/graph.json","events_json":"https://pith.science/api/pith-number/5LZ3MKQNN4TQVNWCBTD5ISXVXH/events.json","paper":"https://pith.science/paper/5LZ3MKQN"},"agent_actions":{"view_html":"https://pith.science/pith/5LZ3MKQNN4TQVNWCBTD5ISXVXH","download_json":"https://pith.science/pith/5LZ3MKQNN4TQVNWCBTD5ISXVXH.json","view_paper":"https://pith.science/paper/5LZ3MKQN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1508.02227&json=true","fetch_graph":"https://pith.science/api/pith-number/5LZ3MKQNN4TQVNWCBTD5ISXVXH/graph.json","fetch_events":"https://pith.science/api/pith-number/5LZ3MKQNN4TQVNWCBTD5ISXVXH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5LZ3MKQNN4TQVNWCBTD5ISXVXH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5LZ3MKQNN4TQVNWCBTD5ISXVXH/action/storage_attestation","attest_author":"https://pith.science/pith/5LZ3MKQNN4TQVNWCBTD5ISXVXH/action/author_attestation","sign_citation":"https://pith.science/pith/5LZ3MKQNN4TQVNWCBTD5ISXVXH/action/citation_signature","submit_replication":"https://pith.science/pith/5LZ3MKQNN4TQVNWCBTD5ISXVXH/action/replication_record"}},"created_at":"2026-05-18T01:35:33.820650+00:00","updated_at":"2026-05-18T01:35:33.820650+00:00"}