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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."},"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":"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.02780","created_at":"2026-05-18T00:16:56.839188+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.02780v4","created_at":"2026-05-18T00:16:56.839188+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.02780","created_at":"2026-05-18T00:16:56.839188+00:00"},{"alias_kind":"pith_short_12","alias_value":"V4I44OGJNUOH","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_16","alias_value":"V4I44OGJNUOHXQXJ","created_at":"2026-05-18T12:30:46.583412+00:00"},{"alias_kind":"pith_short_8","alias_value":"V4I44OGJ","created_at":"2026-05-18T12:30:46.583412+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/V4I44OGJNUOHXQXJK5UAOGFSFJ","json":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ.json","graph_json":"https://pith.science/api/pith-number/V4I44OGJNUOHXQXJK5UAOGFSFJ/graph.json","events_json":"https://pith.science/api/pith-number/V4I44OGJNUOHXQXJK5UAOGFSFJ/events.json","paper":"https://pith.science/paper/V4I44OGJ"},"agent_actions":{"view_html":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ","download_json":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ.json","view_paper":"https://pith.science/paper/V4I44OGJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.02780&json=true","fetch_graph":"https://pith.science/api/pith-number/V4I44OGJNUOHXQXJK5UAOGFSFJ/graph.json","fetch_events":"https://pith.science/api/pith-number/V4I44OGJNUOHXQXJK5UAOGFSFJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/action/storage_attestation","attest_author":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/action/author_attestation","sign_citation":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/action/citation_signature","submit_replication":"https://pith.science/pith/V4I44OGJNUOHXQXJK5UAOGFSFJ/action/replication_record"}},"created_at":"2026-05-18T00:16:56.839188+00:00","updated_at":"2026-05-18T00:16:56.839188+00:00"}