{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1994:ZGTS2R4PWX4BQ5ZQW67MM5DVMI","short_pith_number":"pith:ZGTS2R4P","canonical_record":{"source":{"id":"math/9412211","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1994-12-05T17:46:25Z","cross_cats_sorted":[],"title_canon_sha256":"8be28f407ff4a31cf63a6cce973a104c6e219d081ac3ae9500e98fa7a4c66a8a","abstract_canon_sha256":"8da539d8d5cdb6780886b23dbb0d2f68748998c70e5974f084b3b61379c78dd5"},"schema_version":"1.0"},"canonical_sha256":"c9a72d478fb5f8187730b7bec67475620e6b66001ad6effdc0b9319c9d7927c9","source":{"kind":"arxiv","id":"math/9412211","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9412211","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/9412211v1","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9412211","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"pith_short_12","alias_value":"ZGTS2R4PWX4B","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"ZGTS2R4PWX4BQ5ZQ","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"ZGTS2R4P","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1994:ZGTS2R4PWX4BQ5ZQW67MM5DVMI","target":"record","payload":{"canonical_record":{"source":{"id":"math/9412211","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1994-12-05T17:46:25Z","cross_cats_sorted":[],"title_canon_sha256":"8be28f407ff4a31cf63a6cce973a104c6e219d081ac3ae9500e98fa7a4c66a8a","abstract_canon_sha256":"8da539d8d5cdb6780886b23dbb0d2f68748998c70e5974f084b3b61379c78dd5"},"schema_version":"1.0"},"canonical_sha256":"c9a72d478fb5f8187730b7bec67475620e6b66001ad6effdc0b9319c9d7927c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:50.769009Z","signature_b64":"MrrRjq+TAtyxJcBbiWxQqqTK8TfObjRw+QsJ+4+qPzxYvoP6TTDY9hP9Zs25IisoeI+q513dv8XvQ0VtjtCiAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9a72d478fb5f8187730b7bec67475620e6b66001ad6effdc0b9319c9d7927c9","last_reissued_at":"2026-05-18T01:05:50.768404Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:50.768404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9412211","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8YQTN3m1BnXhIzY0p88NzeW0wLuZ+HYrvQXsnrL6csqLA1XBpk+2iYUialShTfl7y0anf9d42R9y1oONMsrMCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:54:41.461919Z"},"content_sha256":"a31db6f9cb24e57d4be272fee5823c75e0566bb09fd400fc5f7247bf191931af","schema_version":"1.0","event_id":"sha256:a31db6f9cb24e57d4be272fee5823c75e0566bb09fd400fc5f7247bf191931af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1994:ZGTS2R4PWX4BQ5ZQW67MM5DVMI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On Convergence of Conditional Expectation Operators","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"C. Bryan Dawson","submitted_at":"1994-12-05T17:46:25Z","abstract_excerpt":"Given an operator $T:U_X(\\Sigma)\\to Y$ or ${T:U(\\Sigma)\\to Y$, one may consider the net of conditional expectation operators $(T_\\pi)$ directed by refinement of the partitions $\\pi$. It has been shown previously that $(T_\\pi)$ does not always converge to $T$. This paper gives several conditions under which this convergence does occur, including complete characterizations when $X={\\bold R}$ or when $X\\sp *$ has the Radon-Nikod\\'ym property."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9412211","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:05:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"E6OjkWSkJ0bcr9EO8k703AZfJm+XtgxX43XN/zJz2sdmtRmZTSmv7Ayf6eUDA5UjRS5zZeiTj9ob/QYSEUHsCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T21:54:41.462258Z"},"content_sha256":"aba0e78405c7004e3fc05c3250d5f8bd7f3194401f22ffd54dc362947d899145","schema_version":"1.0","event_id":"sha256:aba0e78405c7004e3fc05c3250d5f8bd7f3194401f22ffd54dc362947d899145"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZGTS2R4PWX4BQ5ZQW67MM5DVMI/bundle.json","state_url":"https://pith.science/pith/ZGTS2R4PWX4BQ5ZQW67MM5DVMI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZGTS2R4PWX4BQ5ZQW67MM5DVMI/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":"2026-06-23T21:54:41Z","links":{"resolver":"https://pith.science/pith/ZGTS2R4PWX4BQ5ZQW67MM5DVMI","bundle":"https://pith.science/pith/ZGTS2R4PWX4BQ5ZQW67MM5DVMI/bundle.json","state":"https://pith.science/pith/ZGTS2R4PWX4BQ5ZQW67MM5DVMI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZGTS2R4PWX4BQ5ZQW67MM5DVMI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1994:ZGTS2R4PWX4BQ5ZQW67MM5DVMI","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":"8da539d8d5cdb6780886b23dbb0d2f68748998c70e5974f084b3b61379c78dd5","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1994-12-05T17:46:25Z","title_canon_sha256":"8be28f407ff4a31cf63a6cce973a104c6e219d081ac3ae9500e98fa7a4c66a8a"},"schema_version":"1.0","source":{"id":"math/9412211","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9412211","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"arxiv_version","alias_value":"math/9412211v1","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9412211","created_at":"2026-05-18T01:05:50Z"},{"alias_kind":"pith_short_12","alias_value":"ZGTS2R4PWX4B","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"ZGTS2R4PWX4BQ5ZQ","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"ZGTS2R4P","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:aba0e78405c7004e3fc05c3250d5f8bd7f3194401f22ffd54dc362947d899145","target":"graph","created_at":"2026-05-18T01:05:50Z","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":"Given an operator $T:U_X(\\Sigma)\\to Y$ or ${T:U(\\Sigma)\\to Y$, one may consider the net of conditional expectation operators $(T_\\pi)$ directed by refinement of the partitions $\\pi$. It has been shown previously that $(T_\\pi)$ does not always converge to $T$. This paper gives several conditions under which this convergence does occur, including complete characterizations when $X={\\bold R}$ or when $X\\sp *$ has the Radon-Nikod\\'ym property.","authors_text":"C. Bryan Dawson","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"1994-12-05T17:46:25Z","title":"On Convergence of Conditional Expectation Operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9412211","kind":"arxiv","version":1},"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:a31db6f9cb24e57d4be272fee5823c75e0566bb09fd400fc5f7247bf191931af","target":"record","created_at":"2026-05-18T01:05:50Z","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":"8da539d8d5cdb6780886b23dbb0d2f68748998c70e5974f084b3b61379c78dd5","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1994-12-05T17:46:25Z","title_canon_sha256":"8be28f407ff4a31cf63a6cce973a104c6e219d081ac3ae9500e98fa7a4c66a8a"},"schema_version":"1.0","source":{"id":"math/9412211","kind":"arxiv","version":1}},"canonical_sha256":"c9a72d478fb5f8187730b7bec67475620e6b66001ad6effdc0b9319c9d7927c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c9a72d478fb5f8187730b7bec67475620e6b66001ad6effdc0b9319c9d7927c9","first_computed_at":"2026-05-18T01:05:50.768404Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:50.768404Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MrrRjq+TAtyxJcBbiWxQqqTK8TfObjRw+QsJ+4+qPzxYvoP6TTDY9hP9Zs25IisoeI+q513dv8XvQ0VtjtCiAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:50.769009Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9412211","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a31db6f9cb24e57d4be272fee5823c75e0566bb09fd400fc5f7247bf191931af","sha256:aba0e78405c7004e3fc05c3250d5f8bd7f3194401f22ffd54dc362947d899145"],"state_sha256":"1a4d1c0113e1b38fd8c1e9e0cba18296689260f045549090fed0a42672a658a8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OYytXqXTlQ54Pqz/faBNUd4Tytyi4IxeUYzpQI70TYLFIJgW1zi0PnVvJBhy9z2qpYBW1PwbApQB9IYLtxBgCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T21:54:41.464115Z","bundle_sha256":"3cb3c02a23a813c84626752d8784e9d61627a8081a9f619603dfe6e49044617c"}}