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We prove Fubini's theorem in the discrete setting as follows. If the multiple series (*) converges regularly, then its sum in Pringsheim's sense can be computed by successive summation.\n  We introduce and investigate the regular convergence of the $m$-multiple integral $$\\int^\\infty_0 \\int^\\infty_0...\\int^\\infty_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m,\\leqno(**)$$ where $f: \\ba"},"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":"1112.4950","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-12-21T08:40:15Z","cross_cats_sorted":[],"title_canon_sha256":"9c12740f7a87b528efb8d004864503d4a3faccc994f4519336f482eefc8af096","abstract_canon_sha256":"f607d63ed1db052e8f3d2816f78bd04940adea49efa5b364fe358ba07aab39e4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:56.723296Z","signature_b64":"HjZJZiT8M6WZo7APsiRJHNMjYGX172aSFKjUk0/Ndj+6hK522QTf/y2pRsLa/nN40+NNQNSYmfAY+iZWMO8DCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"caa169eaf5394ea5747c103d54fa4f8a03a08efe20b65e7cd6254a7c7aadf13c","last_reissued_at":"2026-05-18T04:05:56.722701Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:56.722701Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the regular convergence of multiple series of numbers and multiple integrals of locally integrable functions over $\\bar{\\R}^m_+$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ferenc Moricz","submitted_at":"2011-12-21T08:40:15Z","abstract_excerpt":"We investigate the regular convergence of the $m$-multiple series $$\\sum^\\infty_{j_1=0} \\sum^\\infty_{j_2=0}...\\sum^\\infty_{j_m=0} \\ c_{j_1, j_2,..., j_m}\\leqno(*)$$ of complex numbers, where $m\\ge 2$ is a fixed integer. We prove Fubini's theorem in the discrete setting as follows. If the multiple series (*) converges regularly, then its sum in Pringsheim's sense can be computed by successive summation.\n  We introduce and investigate the regular convergence of the $m$-multiple integral $$\\int^\\infty_0 \\int^\\infty_0...\\int^\\infty_0 f(t_1, t_2,..., t_m) dt_1 dt_2...dt_m,\\leqno(**)$$ where $f: \\ba"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4950","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1112.4950","created_at":"2026-05-18T04:05:56.722799+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.4950v1","created_at":"2026-05-18T04:05:56.722799+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4950","created_at":"2026-05-18T04:05:56.722799+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZKQWT2XVHFHK","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZKQWT2XVHFHKK5D4","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZKQWT2XV","created_at":"2026-05-18T12:26:47.523578+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/ZKQWT2XVHFHKK5D4CA6VJ6SPRI","json":"https://pith.science/pith/ZKQWT2XVHFHKK5D4CA6VJ6SPRI.json","graph_json":"https://pith.science/api/pith-number/ZKQWT2XVHFHKK5D4CA6VJ6SPRI/graph.json","events_json":"https://pith.science/api/pith-number/ZKQWT2XVHFHKK5D4CA6VJ6SPRI/events.json","paper":"https://pith.science/paper/ZKQWT2XV"},"agent_actions":{"view_html":"https://pith.science/pith/ZKQWT2XVHFHKK5D4CA6VJ6SPRI","download_json":"https://pith.science/pith/ZKQWT2XVHFHKK5D4CA6VJ6SPRI.json","view_paper":"https://pith.science/paper/ZKQWT2XV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.4950&json=true","fetch_graph":"https://pith.science/api/pith-number/ZKQWT2XVHFHKK5D4CA6VJ6SPRI/graph.json","fetch_events":"https://pith.science/api/pith-number/ZKQWT2XVHFHKK5D4CA6VJ6SPRI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZKQWT2XVHFHKK5D4CA6VJ6SPRI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZKQWT2XVHFHKK5D4CA6VJ6SPRI/action/storage_attestation","attest_author":"https://pith.science/pith/ZKQWT2XVHFHKK5D4CA6VJ6SPRI/action/author_attestation","sign_citation":"https://pith.science/pith/ZKQWT2XVHFHKK5D4CA6VJ6SPRI/action/citation_signature","submit_replication":"https://pith.science/pith/ZKQWT2XVHFHKK5D4CA6VJ6SPRI/action/replication_record"}},"created_at":"2026-05-18T04:05:56.722799+00:00","updated_at":"2026-05-18T04:05:56.722799+00:00"}