{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:JQST5S2QPRZ6U6UHNAPYWKKJDL","short_pith_number":"pith:JQST5S2Q","schema_version":"1.0","canonical_sha256":"4c253ecb507c73ea7a87681f8b29491af1ccdafd5136cd86e9ed1ac3f7f4c2ae","source":{"kind":"arxiv","id":"1601.00639","version":1},"attestation_state":"computed","paper":{"title":"On the equivalence of probability spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniel Alpay, David Levanony, Palle Jorgensen","submitted_at":"2016-01-04T20:57:43Z","abstract_excerpt":"For a general class of Gaussian processes $W$, indexed by a sigma-algebra $\\mathscr F$ of a general measure space $(M,\\mathscr F, \\sigma)$, we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering $\\sigma(A)$, for $A\\in\\mathscr F$, as a quadratic variation of $W$ over $A$. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: $(i)$ a computation of generalized Ito-integrals; and $(ii)$ a proof of an explicit, and measure-theoretic equivale"},"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":"1601.00639","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-04T20:57:43Z","cross_cats_sorted":[],"title_canon_sha256":"35dbd428dedc38e67226cfe5c4fb8e49e72193de7b2a871514af58210562ce05","abstract_canon_sha256":"60e483bed56817db88836ba6a4fe71606c6b8a36734f28b2c6374c6e905f82fe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:26.878936Z","signature_b64":"TPkKGzRvV5arN1JsbnPGdOjI/vRlg79fChOy2I7uiJ2SD3/N5I3KnRVABvw7y3HiKPOe8tv/JqKXnjW6beD6Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4c253ecb507c73ea7a87681f8b29491af1ccdafd5136cd86e9ed1ac3f7f4c2ae","last_reissued_at":"2026-05-18T01:23:26.878371Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:26.878371Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the equivalence of probability spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniel Alpay, David Levanony, Palle Jorgensen","submitted_at":"2016-01-04T20:57:43Z","abstract_excerpt":"For a general class of Gaussian processes $W$, indexed by a sigma-algebra $\\mathscr F$ of a general measure space $(M,\\mathscr F, \\sigma)$, we give necessary and sufficient conditions for the validity of a quadratic variation representation for such Gaussian processes, thus recovering $\\sigma(A)$, for $A\\in\\mathscr F$, as a quadratic variation of $W$ over $A$. We further provide a harmonic analysis representation for this general class of processes. We apply these two results to: $(i)$ a computation of generalized Ito-integrals; and $(ii)$ a proof of an explicit, and measure-theoretic equivale"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.00639","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":"1601.00639","created_at":"2026-05-18T01:23:26.878467+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.00639v1","created_at":"2026-05-18T01:23:26.878467+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.00639","created_at":"2026-05-18T01:23:26.878467+00:00"},{"alias_kind":"pith_short_12","alias_value":"JQST5S2QPRZ6","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"JQST5S2QPRZ6U6UH","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"JQST5S2Q","created_at":"2026-05-18T12:30:25.849896+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/JQST5S2QPRZ6U6UHNAPYWKKJDL","json":"https://pith.science/pith/JQST5S2QPRZ6U6UHNAPYWKKJDL.json","graph_json":"https://pith.science/api/pith-number/JQST5S2QPRZ6U6UHNAPYWKKJDL/graph.json","events_json":"https://pith.science/api/pith-number/JQST5S2QPRZ6U6UHNAPYWKKJDL/events.json","paper":"https://pith.science/paper/JQST5S2Q"},"agent_actions":{"view_html":"https://pith.science/pith/JQST5S2QPRZ6U6UHNAPYWKKJDL","download_json":"https://pith.science/pith/JQST5S2QPRZ6U6UHNAPYWKKJDL.json","view_paper":"https://pith.science/paper/JQST5S2Q","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.00639&json=true","fetch_graph":"https://pith.science/api/pith-number/JQST5S2QPRZ6U6UHNAPYWKKJDL/graph.json","fetch_events":"https://pith.science/api/pith-number/JQST5S2QPRZ6U6UHNAPYWKKJDL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JQST5S2QPRZ6U6UHNAPYWKKJDL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JQST5S2QPRZ6U6UHNAPYWKKJDL/action/storage_attestation","attest_author":"https://pith.science/pith/JQST5S2QPRZ6U6UHNAPYWKKJDL/action/author_attestation","sign_citation":"https://pith.science/pith/JQST5S2QPRZ6U6UHNAPYWKKJDL/action/citation_signature","submit_replication":"https://pith.science/pith/JQST5S2QPRZ6U6UHNAPYWKKJDL/action/replication_record"}},"created_at":"2026-05-18T01:23:26.878467+00:00","updated_at":"2026-05-18T01:23:26.878467+00:00"}