{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:VSTBIK32NOYF2KGHHLWC4LGYED","short_pith_number":"pith:VSTBIK32","canonical_record":{"source":{"id":"2011.00412","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-11-01T03:55:23Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"087a304e168e4fce927a11537f1df97e526713114671b7650981a82b86331b48","abstract_canon_sha256":"7f6a3e7b227a25a63563199cee3c6a88dba1cbcf2110242e424adff4c58a896f"},"schema_version":"1.0"},"canonical_sha256":"aca6142b7a6bb05d28c73aec2e2cd820d1c72361f0a7acfe89620fe56f383054","source":{"kind":"arxiv","id":"2011.00412","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.00412","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"arxiv_version","alias_value":"2011.00412v3","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.00412","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"pith_short_12","alias_value":"VSTBIK32NOYF","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"pith_short_16","alias_value":"VSTBIK32NOYF2KGH","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"pith_short_8","alias_value":"VSTBIK32","created_at":"2026-07-05T05:36:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:VSTBIK32NOYF2KGHHLWC4LGYED","target":"record","payload":{"canonical_record":{"source":{"id":"2011.00412","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-11-01T03:55:23Z","cross_cats_sorted":["math.CT"],"title_canon_sha256":"087a304e168e4fce927a11537f1df97e526713114671b7650981a82b86331b48","abstract_canon_sha256":"7f6a3e7b227a25a63563199cee3c6a88dba1cbcf2110242e424adff4c58a896f"},"schema_version":"1.0"},"canonical_sha256":"aca6142b7a6bb05d28c73aec2e2cd820d1c72361f0a7acfe89620fe56f383054","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:36:43.843496Z","signature_b64":"Rd1pbne96eUOzfbOY9M4QSu7/YAaK3CLAoICqkWSPHl60Qg9IdkkiBy6DP261crQgKhlnpRYRTkqcLhR0qzWCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aca6142b7a6bb05d28c73aec2e2cd820d1c72361f0a7acfe89620fe56f383054","last_reissued_at":"2026-07-05T05:36:43.842979Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:36:43.842979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2011.00412","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T05:36:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ztNJq3MbtRDdRFS0xeIqecJdgH+a1Nw/vTelj0M19JwtIGGrlGMD0Fe1kjAz101KH1X91/uZAbdjodSGDB8DDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T07:15:48.014130Z"},"content_sha256":"84e38e31d5a06b5ffe3b5f407f06781963f4b2d1ac63afa0d89aab02953b79c3","schema_version":"1.0","event_id":"sha256:84e38e31d5a06b5ffe3b5f407f06781963f4b2d1ac63afa0d89aab02953b79c3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:VSTBIK32NOYF2KGHHLWC4LGYED","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A categorical derivation of Lebesgue integration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CT"],"primary_cat":"math.FA","authors_text":"Tom Leinster","submitted_at":"2020-11-01T03:55:23Z","abstract_excerpt":"We identify simple universal properties that uniquely characterize the Lebesgue $L^p$ spaces. There are two main theorems. The first states that the Banach space $L^p[0, 1]$, equipped with a small amount of extra structure, is initial as such. The second states that the $L^p$ functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. Using the universal properties, we develop some of the basic elements of integration theory. We also state"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.00412","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2011.00412/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T05:36:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PqL+2B0c+cqwx16t+PrcF1186ncY2TgGM6n1RnsjTqQ0n53puMwtDRsCawLqADpUcTw+cAyEmpC/AUITvIaQBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-20T07:15:48.014767Z"},"content_sha256":"4e647f700a3148bc2e9806425a5b8905d4c97d9963ab0936782c643ee107e78e","schema_version":"1.0","event_id":"sha256:4e647f700a3148bc2e9806425a5b8905d4c97d9963ab0936782c643ee107e78e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VSTBIK32NOYF2KGHHLWC4LGYED/bundle.json","state_url":"https://pith.science/pith/VSTBIK32NOYF2KGHHLWC4LGYED/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VSTBIK32NOYF2KGHHLWC4LGYED/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":"2026-08-20T07:15:48Z","links":{"resolver":"https://pith.science/pith/VSTBIK32NOYF2KGHHLWC4LGYED","bundle":"https://pith.science/pith/VSTBIK32NOYF2KGHHLWC4LGYED/bundle.json","state":"https://pith.science/pith/VSTBIK32NOYF2KGHHLWC4LGYED/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VSTBIK32NOYF2KGHHLWC4LGYED/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:VSTBIK32NOYF2KGHHLWC4LGYED","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":"7f6a3e7b227a25a63563199cee3c6a88dba1cbcf2110242e424adff4c58a896f","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-11-01T03:55:23Z","title_canon_sha256":"087a304e168e4fce927a11537f1df97e526713114671b7650981a82b86331b48"},"schema_version":"1.0","source":{"id":"2011.00412","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2011.00412","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"arxiv_version","alias_value":"2011.00412v3","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2011.00412","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"pith_short_12","alias_value":"VSTBIK32NOYF","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"pith_short_16","alias_value":"VSTBIK32NOYF2KGH","created_at":"2026-07-05T05:36:43Z"},{"alias_kind":"pith_short_8","alias_value":"VSTBIK32","created_at":"2026-07-05T05:36:43Z"}],"graph_snapshots":[{"event_id":"sha256:4e647f700a3148bc2e9806425a5b8905d4c97d9963ab0936782c643ee107e78e","target":"graph","created_at":"2026-07-05T05:36:43Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2011.00412/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We identify simple universal properties that uniquely characterize the Lebesgue $L^p$ spaces. There are two main theorems. The first states that the Banach space $L^p[0, 1]$, equipped with a small amount of extra structure, is initial as such. The second states that the $L^p$ functor on finite measure spaces, again with some extra structure, is also initial as such. In both cases, the universal characterization of the integrable functions produces a unique characterization of integration. Using the universal properties, we develop some of the basic elements of integration theory. We also state","authors_text":"Tom Leinster","cross_cats":["math.CT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-11-01T03:55:23Z","title":"A categorical derivation of Lebesgue integration"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2011.00412","kind":"arxiv","version":3},"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":"sha256:84e38e31d5a06b5ffe3b5f407f06781963f4b2d1ac63afa0d89aab02953b79c3","target":"record","created_at":"2026-07-05T05:36:43Z","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":"7f6a3e7b227a25a63563199cee3c6a88dba1cbcf2110242e424adff4c58a896f","cross_cats_sorted":["math.CT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2020-11-01T03:55:23Z","title_canon_sha256":"087a304e168e4fce927a11537f1df97e526713114671b7650981a82b86331b48"},"schema_version":"1.0","source":{"id":"2011.00412","kind":"arxiv","version":3}},"canonical_sha256":"aca6142b7a6bb05d28c73aec2e2cd820d1c72361f0a7acfe89620fe56f383054","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aca6142b7a6bb05d28c73aec2e2cd820d1c72361f0a7acfe89620fe56f383054","first_computed_at":"2026-07-05T05:36:43.842979Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:36:43.842979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Rd1pbne96eUOzfbOY9M4QSu7/YAaK3CLAoICqkWSPHl60Qg9IdkkiBy6DP261crQgKhlnpRYRTkqcLhR0qzWCw==","signature_status":"signed_v1","signed_at":"2026-07-05T05:36:43.843496Z","signed_message":"canonical_sha256_bytes"},"source_id":"2011.00412","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:84e38e31d5a06b5ffe3b5f407f06781963f4b2d1ac63afa0d89aab02953b79c3","sha256:4e647f700a3148bc2e9806425a5b8905d4c97d9963ab0936782c643ee107e78e"],"state_sha256":"3133e45392b5c8d337b56693379767fe42f85154258951ea35fd39790d685d8a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vkZrip1cGCC53nIYHYFD149yXeTdTesnShuocPzp9I9REwn+pI5o4+8zHzHuacX53Af+/gtNbYXoWILDQ72EAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-20T07:15:48.020284Z","bundle_sha256":"8ebf58acf3abc83e1b039cf03b0fdacb1b910498e5f2205fd05f6be1dbeadaec"}}