{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:VUQCFGZLK3WE7BQEUV336EOEJI","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":"163018b3ab98778c49521c52ad888eb4fe5f8f0f0a584996c17a4e62741de7f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-23T13:22:00Z","title_canon_sha256":"0c15342cb2e09a357c9a98ba9c23ccf235a99b8b5be616757d3e9bbbe00fc5c5"},"schema_version":"1.0","source":{"id":"1509.06965","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.06965","created_at":"2026-05-18T01:18:14Z"},{"alias_kind":"arxiv_version","alias_value":"1509.06965v1","created_at":"2026-05-18T01:18:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.06965","created_at":"2026-05-18T01:18:14Z"},{"alias_kind":"pith_short_12","alias_value":"VUQCFGZLK3WE","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"VUQCFGZLK3WE7BQE","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"VUQCFGZL","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:fb78639d1ec8d27bb1c928a477b9ff553ee5bc29954edc69bf9f99b73bb2f044","target":"graph","created_at":"2026-05-18T01:18:14Z","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 formulate the saturation property for vector measures in lcHs as a nonseparability condition on the derived Boolean $\\sigma$-algebras by drawing on the topological structure of vector measure algebras. We exploit a Pettis-like notion of vector integration in lcHs, the Bourbaki--Kluv\\'anek--Lewis integral, to derive an exact version of the Lyapunov convexity theorem in lcHs without the BDS property. We apply our Lyapunov convexity theorem to the bang-bang principle in Lyapunov control systems in lcHs to provide a further characterization of the saturation property.","authors_text":"M. Ali Khan, Nobusumi Sagara","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-23T13:22:00Z","title":"Maharam-Types and Lyapunov's Theorem for Vector Measures on Locally Convex Spaces without Control Measures"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06965","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":"sha256:86cb850ecbcd135166868ac6dafac5ebda1f084f6cf60b423473f21f61a7567f","target":"record","created_at":"2026-05-18T01:18:14Z","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":"163018b3ab98778c49521c52ad888eb4fe5f8f0f0a584996c17a4e62741de7f4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2015-09-23T13:22:00Z","title_canon_sha256":"0c15342cb2e09a357c9a98ba9c23ccf235a99b8b5be616757d3e9bbbe00fc5c5"},"schema_version":"1.0","source":{"id":"1509.06965","kind":"arxiv","version":1}},"canonical_sha256":"ad20229b2b56ec4f8604a577bf11c44a28052d69c243db75dd98247237a493e1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ad20229b2b56ec4f8604a577bf11c44a28052d69c243db75dd98247237a493e1","first_computed_at":"2026-05-18T01:18:14.301600Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:18:14.301600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ieXBtmxx3vRveS7NPy+/mwPB1wWgH7LQisQkDtwJ0QyafU26UtaceUGRHBbBEnbHJZSYy37SfbG7M7jXQZEPAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:18:14.302274Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.06965","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:86cb850ecbcd135166868ac6dafac5ebda1f084f6cf60b423473f21f61a7567f","sha256:fb78639d1ec8d27bb1c928a477b9ff553ee5bc29954edc69bf9f99b73bb2f044"],"state_sha256":"c5a40d953dec3e09aff44d1e543fd6a625934e511ee2312d8563a8be3861848a"}