{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:L7XOKHVOI6MHDTUROJ5K36EPY3","short_pith_number":"pith:L7XOKHVO","schema_version":"1.0","canonical_sha256":"5feee51eae479871ce91727aadf88fc6c72d5a9f964236b0c2a5dc976c28462f","source":{"kind":"arxiv","id":"1011.0212","version":1},"attestation_state":"computed","paper":{"title":"Continuity of Translation Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Justin R. Peters, Krishna B. Athreya","submitted_at":"2010-10-31T21:33:15Z","abstract_excerpt":"For a Radon measure $\\mu$ on $\\bbR,$ we show that $L^{\\infty}(\\mu)$ is invariant under the group of translation operators $T_t(f)(x) = {$f(x-t)$}\\ (t \\in \\bbR)$ if and only if $\\mu$ is equivalent to Lebesgue measure $m$. We also give necessary and sufficient conditions for $L^p(\\mu),\\1 \\leq p < \\infty,$ to be invariant under the group $\\{T_t\\}$ in terms of the Radon-Nikodym derivative w.r.t. $m$."},"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":"1011.0212","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2010-10-31T21:33:15Z","cross_cats_sorted":[],"title_canon_sha256":"96249de1a65c8b09f926decb0f202118c0bf397aa8a30d4e1633bb23db991694","abstract_canon_sha256":"12882bbe7b0f8a0be20338723a7947b034c52235983096bb3bf593659f8561d7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:37:30.380725Z","signature_b64":"z4V2UFVLPkcLgutliMT22mBerAMdgooHPgOyL/6aWriytoa4PCJAPi6zC5ScGm2z0SptS7tVN4O5UsQb+ZamDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5feee51eae479871ce91727aadf88fc6c72d5a9f964236b0c2a5dc976c28462f","last_reissued_at":"2026-05-18T04:37:30.380076Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:37:30.380076Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Continuity of Translation Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Justin R. Peters, Krishna B. Athreya","submitted_at":"2010-10-31T21:33:15Z","abstract_excerpt":"For a Radon measure $\\mu$ on $\\bbR,$ we show that $L^{\\infty}(\\mu)$ is invariant under the group of translation operators $T_t(f)(x) = {$f(x-t)$}\\ (t \\in \\bbR)$ if and only if $\\mu$ is equivalent to Lebesgue measure $m$. We also give necessary and sufficient conditions for $L^p(\\mu),\\1 \\leq p < \\infty,$ to be invariant under the group $\\{T_t\\}$ in terms of the Radon-Nikodym derivative w.r.t. $m$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0212","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":"1011.0212","created_at":"2026-05-18T04:37:30.380160+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.0212v1","created_at":"2026-05-18T04:37:30.380160+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.0212","created_at":"2026-05-18T04:37:30.380160+00:00"},{"alias_kind":"pith_short_12","alias_value":"L7XOKHVOI6MH","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_16","alias_value":"L7XOKHVOI6MHDTUR","created_at":"2026-05-18T12:26:10.704358+00:00"},{"alias_kind":"pith_short_8","alias_value":"L7XOKHVO","created_at":"2026-05-18T12:26:10.704358+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/L7XOKHVOI6MHDTUROJ5K36EPY3","json":"https://pith.science/pith/L7XOKHVOI6MHDTUROJ5K36EPY3.json","graph_json":"https://pith.science/api/pith-number/L7XOKHVOI6MHDTUROJ5K36EPY3/graph.json","events_json":"https://pith.science/api/pith-number/L7XOKHVOI6MHDTUROJ5K36EPY3/events.json","paper":"https://pith.science/paper/L7XOKHVO"},"agent_actions":{"view_html":"https://pith.science/pith/L7XOKHVOI6MHDTUROJ5K36EPY3","download_json":"https://pith.science/pith/L7XOKHVOI6MHDTUROJ5K36EPY3.json","view_paper":"https://pith.science/paper/L7XOKHVO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.0212&json=true","fetch_graph":"https://pith.science/api/pith-number/L7XOKHVOI6MHDTUROJ5K36EPY3/graph.json","fetch_events":"https://pith.science/api/pith-number/L7XOKHVOI6MHDTUROJ5K36EPY3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/L7XOKHVOI6MHDTUROJ5K36EPY3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/L7XOKHVOI6MHDTUROJ5K36EPY3/action/storage_attestation","attest_author":"https://pith.science/pith/L7XOKHVOI6MHDTUROJ5K36EPY3/action/author_attestation","sign_citation":"https://pith.science/pith/L7XOKHVOI6MHDTUROJ5K36EPY3/action/citation_signature","submit_replication":"https://pith.science/pith/L7XOKHVOI6MHDTUROJ5K36EPY3/action/replication_record"}},"created_at":"2026-05-18T04:37:30.380160+00:00","updated_at":"2026-05-18T04:37:30.380160+00:00"}