{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:V4I44OGJNUOHXQXJK5UAOGFSFJ","short_pith_number":"pith:V4I44OGJ","canonical_record":{"source":{"id":"1606.02780","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-08T23:12:57Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"5cb896e90451588833e4eaa4678f56f9843ed3115a6516ed80526751164f06b9","abstract_canon_sha256":"16a8eb98fc559aca4c4612b942ba85ec7d88f543d58a1313d95d51a014c7c283"},"schema_version":"1.0"},"canonical_sha256":"af11ce38c96d1c7bc2e957680718b22a6f94eabacdc49054c782fe1b6129af9f","source":{"kind":"arxiv","id":"1606.02780","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02780","created_at":"2026-05-18T00:16:56Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02780v4","created_at":"2026-05-18T00:16:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02780","created_at":"2026-05-18T00:16:56Z"},{"alias_kind":"pith_short_12","alias_value":"V4I44OGJNUOH","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"V4I44OGJNUOHXQXJ","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"V4I44OGJ","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:V4I44OGJNUOHXQXJK5UAOGFSFJ","target":"record","payload":{"canonical_record":{"source":{"id":"1606.02780","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-08T23:12:57Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"5cb896e90451588833e4eaa4678f56f9843ed3115a6516ed80526751164f06b9","abstract_canon_sha256":"16a8eb98fc559aca4c4612b942ba85ec7d88f543d58a1313d95d51a014c7c283"},"schema_version":"1.0"},"canonical_sha256":"af11ce38c96d1c7bc2e957680718b22a6f94eabacdc49054c782fe1b6129af9f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:56.839784Z","signature_b64":"5IH2QAysqNeBBNYTjZUaKO4hWCGiocXLRHQhS9WdGyAKb0osLlttgut8RL55yVIpdXREVr2mBbrE2zgqTEFVBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af11ce38c96d1c7bc2e957680718b22a6f94eabacdc49054c782fe1b6129af9f","last_reissued_at":"2026-05-18T00:16:56.839092Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:56.839092Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.02780","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:16:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d/OR+SmsCJGYICZ7XaleYVtDTSRmzEly9B76mJ3nJuBCzBZiPVgBd5N2Es5nGUmhK+1vcJN0av6dN7VxmencDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T13:35:05.997736Z"},"content_sha256":"057f3fd3769a87fb6895ad68e1e4fa3fb2bc5db35cf83f7a798f2bd53caf8771","schema_version":"1.0","event_id":"sha256:057f3fd3769a87fb6895ad68e1e4fa3fb2bc5db35cf83f7a798f2bd53caf8771"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:V4I44OGJNUOHXQXJK5UAOGFSFJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On conditional expectations in L^p(mu;L^q(nu;X))","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Jan van Neerven, Qi Lu","submitted_at":"2016-06-08T23:12:57Z","abstract_excerpt":"Let $(A,\\mathscr{A},\\mu)$ and $(B,\\mathscr{B},\\nu)$ be probability spaces, let $\\mathscr{F}$ be a sub-$\\sigma$-algebra of the product $\\sigma$-algebra $\\mathscr{A}\\times\\mathscr{B}$, let $X$ be a Banach space, and let $1< p,q< \\infty$. We obtain necessary and sufficient conditions in order that the conditional expectation with respect to $\\mathscr{F}$ defines a bounded linear operator from $L^p(\\mu;L^q(\\nu;X))$ onto $L^p_{\\mathscr{F}}(\\mu;L^q(\\nu;X))$, the closed subspace in $L^p(\\mu;L^q(\\nu;X))$ of all functions having a strongly $\\mathscr{F}$-measurable representative."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02780","kind":"arxiv","version":4},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:16:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B1vW9YZVq/jPePU43Kbw3YoMfZBvVuUzyOd64evR15AokKsGyUM4X8IxPurJri18h8a4TpX9fwM/w7dPGmWQCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T13:35:05.998079Z"},"content_sha256":"7995cb2996d91c5d5727a8432e3f9e6cb206872cb550ccff91614927e37dfb22","schema_version":"1.0","event_id":"sha256:7995cb2996d91c5d5727a8432e3f9e6cb206872cb550ccff91614927e37dfb22"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/bundle.json","state_url":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-04T13:35:05Z","links":{"resolver":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ","bundle":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/bundle.json","state":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:V4I44OGJNUOHXQXJK5UAOGFSFJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"16a8eb98fc559aca4c4612b942ba85ec7d88f543d58a1313d95d51a014c7c283","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-08T23:12:57Z","title_canon_sha256":"5cb896e90451588833e4eaa4678f56f9843ed3115a6516ed80526751164f06b9"},"schema_version":"1.0","source":{"id":"1606.02780","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.02780","created_at":"2026-05-18T00:16:56Z"},{"alias_kind":"arxiv_version","alias_value":"1606.02780v4","created_at":"2026-05-18T00:16:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02780","created_at":"2026-05-18T00:16:56Z"},{"alias_kind":"pith_short_12","alias_value":"V4I44OGJNUOH","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"V4I44OGJNUOHXQXJ","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"V4I44OGJ","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:7995cb2996d91c5d5727a8432e3f9e6cb206872cb550ccff91614927e37dfb22","target":"graph","created_at":"2026-05-18T00:16:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $(A,\\mathscr{A},\\mu)$ and $(B,\\mathscr{B},\\nu)$ be probability spaces, let $\\mathscr{F}$ be a sub-$\\sigma$-algebra of the product $\\sigma$-algebra $\\mathscr{A}\\times\\mathscr{B}$, let $X$ be a Banach space, and let $1< p,q< \\infty$. We obtain necessary and sufficient conditions in order that the conditional expectation with respect to $\\mathscr{F}$ defines a bounded linear operator from $L^p(\\mu;L^q(\\nu;X))$ onto $L^p_{\\mathscr{F}}(\\mu;L^q(\\nu;X))$, the closed subspace in $L^p(\\mu;L^q(\\nu;X))$ of all functions having a strongly $\\mathscr{F}$-measurable representative.","authors_text":"Jan van Neerven, Qi Lu","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-08T23:12:57Z","title":"On conditional expectations in L^p(mu;L^q(nu;X))"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.02780","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:057f3fd3769a87fb6895ad68e1e4fa3fb2bc5db35cf83f7a798f2bd53caf8771","target":"record","created_at":"2026-05-18T00:16:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"16a8eb98fc559aca4c4612b942ba85ec7d88f543d58a1313d95d51a014c7c283","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-06-08T23:12:57Z","title_canon_sha256":"5cb896e90451588833e4eaa4678f56f9843ed3115a6516ed80526751164f06b9"},"schema_version":"1.0","source":{"id":"1606.02780","kind":"arxiv","version":4}},"canonical_sha256":"af11ce38c96d1c7bc2e957680718b22a6f94eabacdc49054c782fe1b6129af9f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af11ce38c96d1c7bc2e957680718b22a6f94eabacdc49054c782fe1b6129af9f","first_computed_at":"2026-05-18T00:16:56.839092Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:56.839092Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5IH2QAysqNeBBNYTjZUaKO4hWCGiocXLRHQhS9WdGyAKb0osLlttgut8RL55yVIpdXREVr2mBbrE2zgqTEFVBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:56.839784Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.02780","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:057f3fd3769a87fb6895ad68e1e4fa3fb2bc5db35cf83f7a798f2bd53caf8771","sha256:7995cb2996d91c5d5727a8432e3f9e6cb206872cb550ccff91614927e37dfb22"],"state_sha256":"b2203cecd8eb40a6b81c5a42052b9306676448f25e65673982621c3348c89e1f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6bUrxlj7o6+A4qJV9E9D5ZV7ROXfSMNRaiWypGX2+Zd0bNCvnCPrRSsQASGcbBo5JzfHPNVGWTMHiYYk2zpVDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T13:35:06.000043Z","bundle_sha256":"bf831bd517fc80c13d1b6227c5371d93beacdb1f0b72e0b14b84f05d7d4b26c2"}}