{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ZQBV3TQHEDDMVB4XHNYRBEPPLL","short_pith_number":"pith:ZQBV3TQH","canonical_record":{"source":{"id":"1607.07482","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-25T21:17:27Z","cross_cats_sorted":[],"title_canon_sha256":"201322a9557c234b1e91adc1d93d89469330c63ad1349b988471e94f509036f7","abstract_canon_sha256":"9884df0922e084faf212adc6e394640786d69d4676a3c6ddfa4e762af2ff97f1"},"schema_version":"1.0"},"canonical_sha256":"cc035dce0720c6ca87973b711091ef5ac3da88cb91fcb02cfa6ad58d8c966eae","source":{"kind":"arxiv","id":"1607.07482","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.07482","created_at":"2026-05-18T01:10:28Z"},{"alias_kind":"arxiv_version","alias_value":"1607.07482v1","created_at":"2026-05-18T01:10:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07482","created_at":"2026-05-18T01:10:28Z"},{"alias_kind":"pith_short_12","alias_value":"ZQBV3TQHEDDM","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZQBV3TQHEDDMVB4X","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZQBV3TQH","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ZQBV3TQHEDDMVB4XHNYRBEPPLL","target":"record","payload":{"canonical_record":{"source":{"id":"1607.07482","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-25T21:17:27Z","cross_cats_sorted":[],"title_canon_sha256":"201322a9557c234b1e91adc1d93d89469330c63ad1349b988471e94f509036f7","abstract_canon_sha256":"9884df0922e084faf212adc6e394640786d69d4676a3c6ddfa4e762af2ff97f1"},"schema_version":"1.0"},"canonical_sha256":"cc035dce0720c6ca87973b711091ef5ac3da88cb91fcb02cfa6ad58d8c966eae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:28.488917Z","signature_b64":"JLkSt9AMwFSZXb0KLSVqWUM8nWFY3toU5G2/Cr/i/NMzoDzy8X3ZFcI4GYK2YmdwpYmluxrZvld6K2PF9YVuAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cc035dce0720c6ca87973b711091ef5ac3da88cb91fcb02cfa6ad58d8c966eae","last_reissued_at":"2026-05-18T01:10:28.488546Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:28.488546Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.07482","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":"2026-05-18T01:10:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QF5M4be2xUTDE7aehdDR3NK4j2MQkXhgKwn2y6artCPtLNpzDygEyaWXSNEHiXunmAHEqzmgiLviDXlTK6ZTBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T05:18:30.747825Z"},"content_sha256":"dba5a4750f605d6d63077229c0ef4ef533b67314c2c62dd705570fa792311c2b","schema_version":"1.0","event_id":"sha256:dba5a4750f605d6d63077229c0ef4ef533b67314c2c62dd705570fa792311c2b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ZQBV3TQHEDDMVB4XHNYRBEPPLL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"More on measurable algebras and Rademacher systems with applications to analysis of Riesz spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Mikhail Popov","submitted_at":"2016-07-25T21:17:27Z","abstract_excerpt":"We find necessary and sufficient conditions on a family $\\mathcal{R} = (r_i)_{i \\in I}$ in a Boolean algebra $\\mathcal{B}$ under which there exists a unique positive probability measure $\\mu$ on $\\mathcal{B}$ such that $\\mu ( \\bigcap_{k=1}^n \\theta_k r_{i_k} ) = 2^{-n}$ for all finite collections of distinct indices $i_1, \\ldots, i_n \\in I$ and all collections of signs $\\theta_1, \\ldots, \\theta_n \\in \\{-1,1\\}$, where the product $\\theta x$ of a sign $\\theta$ by an element $x \\in \\mathcal{B}$ is defined by setting $1 x = x$ and $-1 x = - x = \\mathbf{1} \\setminus x$. Such a family we call a comp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07482","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":"2026-05-18T01:10:28Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"j73V5jpD4pLIK3QioyzXaZkI+c13pfAMJykVZ63I+y+qVtJrU17neJgWAB8h+kn9rlHljs2A1MCEJNiy+CDMBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T05:18:30.748180Z"},"content_sha256":"6aae8ef84b48759b3249c90ca9e9905c686d6809d8137b6f837758464c089f6e","schema_version":"1.0","event_id":"sha256:6aae8ef84b48759b3249c90ca9e9905c686d6809d8137b6f837758464c089f6e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZQBV3TQHEDDMVB4XHNYRBEPPLL/bundle.json","state_url":"https://pith.science/pith/ZQBV3TQHEDDMVB4XHNYRBEPPLL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZQBV3TQHEDDMVB4XHNYRBEPPLL/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":"2026-06-12T05:18:30Z","links":{"resolver":"https://pith.science/pith/ZQBV3TQHEDDMVB4XHNYRBEPPLL","bundle":"https://pith.science/pith/ZQBV3TQHEDDMVB4XHNYRBEPPLL/bundle.json","state":"https://pith.science/pith/ZQBV3TQHEDDMVB4XHNYRBEPPLL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZQBV3TQHEDDMVB4XHNYRBEPPLL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZQBV3TQHEDDMVB4XHNYRBEPPLL","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":"9884df0922e084faf212adc6e394640786d69d4676a3c6ddfa4e762af2ff97f1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-25T21:17:27Z","title_canon_sha256":"201322a9557c234b1e91adc1d93d89469330c63ad1349b988471e94f509036f7"},"schema_version":"1.0","source":{"id":"1607.07482","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.07482","created_at":"2026-05-18T01:10:28Z"},{"alias_kind":"arxiv_version","alias_value":"1607.07482v1","created_at":"2026-05-18T01:10:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07482","created_at":"2026-05-18T01:10:28Z"},{"alias_kind":"pith_short_12","alias_value":"ZQBV3TQHEDDM","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"ZQBV3TQHEDDMVB4X","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"ZQBV3TQH","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:6aae8ef84b48759b3249c90ca9e9905c686d6809d8137b6f837758464c089f6e","target":"graph","created_at":"2026-05-18T01:10:28Z","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":"We find necessary and sufficient conditions on a family $\\mathcal{R} = (r_i)_{i \\in I}$ in a Boolean algebra $\\mathcal{B}$ under which there exists a unique positive probability measure $\\mu$ on $\\mathcal{B}$ such that $\\mu ( \\bigcap_{k=1}^n \\theta_k r_{i_k} ) = 2^{-n}$ for all finite collections of distinct indices $i_1, \\ldots, i_n \\in I$ and all collections of signs $\\theta_1, \\ldots, \\theta_n \\in \\{-1,1\\}$, where the product $\\theta x$ of a sign $\\theta$ by an element $x \\in \\mathcal{B}$ is defined by setting $1 x = x$ and $-1 x = - x = \\mathbf{1} \\setminus x$. Such a family we call a comp","authors_text":"Mikhail Popov","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-25T21:17:27Z","title":"More on measurable algebras and Rademacher systems with applications to analysis of Riesz spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07482","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:dba5a4750f605d6d63077229c0ef4ef533b67314c2c62dd705570fa792311c2b","target":"record","created_at":"2026-05-18T01:10:28Z","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":"9884df0922e084faf212adc6e394640786d69d4676a3c6ddfa4e762af2ff97f1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-07-25T21:17:27Z","title_canon_sha256":"201322a9557c234b1e91adc1d93d89469330c63ad1349b988471e94f509036f7"},"schema_version":"1.0","source":{"id":"1607.07482","kind":"arxiv","version":1}},"canonical_sha256":"cc035dce0720c6ca87973b711091ef5ac3da88cb91fcb02cfa6ad58d8c966eae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cc035dce0720c6ca87973b711091ef5ac3da88cb91fcb02cfa6ad58d8c966eae","first_computed_at":"2026-05-18T01:10:28.488546Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:10:28.488546Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JLkSt9AMwFSZXb0KLSVqWUM8nWFY3toU5G2/Cr/i/NMzoDzy8X3ZFcI4GYK2YmdwpYmluxrZvld6K2PF9YVuAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:10:28.488917Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.07482","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dba5a4750f605d6d63077229c0ef4ef533b67314c2c62dd705570fa792311c2b","sha256:6aae8ef84b48759b3249c90ca9e9905c686d6809d8137b6f837758464c089f6e"],"state_sha256":"e30bfdf2571a375891483a9056a2ce1382e845cceef500cfaa94729b83f7fef4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ygRVb5xcbtXXS9Mpl3nU+O7DFvhlrCt/l9Am5AGHv4I1YS/unKUVLbIhiKMEcFuwYCHsKO/ozMrqhplc2TKxBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T05:18:30.750306Z","bundle_sha256":"ad400d3114c2003a2d5bcbcc0ee31b301d1c28cb812a7c379900842820f026b9"}}