{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:RYKSTGVQBCVTG3RPFVXWLMWNHB","short_pith_number":"pith:RYKSTGVQ","schema_version":"1.0","canonical_sha256":"8e15299ab008ab336e2f2d6f65b2cd3864d5ec04d91cb91266e74b37d8a4a9c9","source":{"kind":"arxiv","id":"1503.01139","version":3},"attestation_state":"computed","paper":{"title":"On the commutation of generalized means on probability spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Janusz Matkowski, Paolo Leonetti, Salvatore Tringali","submitted_at":"2015-03-03T21:50:27Z","abstract_excerpt":"Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \\mathscr{L}, \\lambda)$ and $(Y, \\mathscr{M}, \\mu)$ be probability spaces in each of which there is at least one measurable set whose measure is strictly between $0$ and $1$.\n  We say that $(f,g)$ is a $(\\lambda, \\mu)$-switch if, for every $\\mathscr{L} \\otimes \\mathscr{M}$-measurable function $h: X \\times Y \\to \\mathbf{R}$ for which $h[X\\times Y]$ is contained in a compact subset of $I$, it holds $$ f^{-1}\\!\\left(\\int_X f\\!\\left(g^{-1}\\!\\left(\\int_Y g \\circ h\\;d\\mu\\right)\\right)d \\lambda\\"},"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":"1503.01139","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-03-03T21:50:27Z","cross_cats_sorted":[],"title_canon_sha256":"4e117aa8a812f6f7f01d7cc92a4c764804ba2cb95a90c870c92c157cbbb02c38","abstract_canon_sha256":"e7c38511b4101517f427f3af998915805b27ce7f10c3489fcb10732f9e1224e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:22.984183Z","signature_b64":"Syid74ROsRIVnX5kDor9+56zhzZGjAs46bsdBfjqiksqJqP7DjozjMgUQXC4C2HSTE72RCwwQLq2mwQ7tRyOBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8e15299ab008ab336e2f2d6f65b2cd3864d5ec04d91cb91266e74b37d8a4a9c9","last_reissued_at":"2026-05-18T01:06:22.983682Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:22.983682Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the commutation of generalized means on probability spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Janusz Matkowski, Paolo Leonetti, Salvatore Tringali","submitted_at":"2015-03-03T21:50:27Z","abstract_excerpt":"Let $f$ and $g$ be real-valued continuous injections defined on a non-empty real interval $I$, and let $(X, \\mathscr{L}, \\lambda)$ and $(Y, \\mathscr{M}, \\mu)$ be probability spaces in each of which there is at least one measurable set whose measure is strictly between $0$ and $1$.\n  We say that $(f,g)$ is a $(\\lambda, \\mu)$-switch if, for every $\\mathscr{L} \\otimes \\mathscr{M}$-measurable function $h: X \\times Y \\to \\mathbf{R}$ for which $h[X\\times Y]$ is contained in a compact subset of $I$, it holds $$ f^{-1}\\!\\left(\\int_X f\\!\\left(g^{-1}\\!\\left(\\int_Y g \\circ h\\;d\\mu\\right)\\right)d \\lambda\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.01139","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":""},"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":"1503.01139","created_at":"2026-05-18T01:06:22.983761+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.01139v3","created_at":"2026-05-18T01:06:22.983761+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.01139","created_at":"2026-05-18T01:06:22.983761+00:00"},{"alias_kind":"pith_short_12","alias_value":"RYKSTGVQBCVT","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_16","alias_value":"RYKSTGVQBCVTG3RP","created_at":"2026-05-18T12:29:39.896362+00:00"},{"alias_kind":"pith_short_8","alias_value":"RYKSTGVQ","created_at":"2026-05-18T12:29:39.896362+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/RYKSTGVQBCVTG3RPFVXWLMWNHB","json":"https://pith.science/pith/RYKSTGVQBCVTG3RPFVXWLMWNHB.json","graph_json":"https://pith.science/api/pith-number/RYKSTGVQBCVTG3RPFVXWLMWNHB/graph.json","events_json":"https://pith.science/api/pith-number/RYKSTGVQBCVTG3RPFVXWLMWNHB/events.json","paper":"https://pith.science/paper/RYKSTGVQ"},"agent_actions":{"view_html":"https://pith.science/pith/RYKSTGVQBCVTG3RPFVXWLMWNHB","download_json":"https://pith.science/pith/RYKSTGVQBCVTG3RPFVXWLMWNHB.json","view_paper":"https://pith.science/paper/RYKSTGVQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.01139&json=true","fetch_graph":"https://pith.science/api/pith-number/RYKSTGVQBCVTG3RPFVXWLMWNHB/graph.json","fetch_events":"https://pith.science/api/pith-number/RYKSTGVQBCVTG3RPFVXWLMWNHB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RYKSTGVQBCVTG3RPFVXWLMWNHB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RYKSTGVQBCVTG3RPFVXWLMWNHB/action/storage_attestation","attest_author":"https://pith.science/pith/RYKSTGVQBCVTG3RPFVXWLMWNHB/action/author_attestation","sign_citation":"https://pith.science/pith/RYKSTGVQBCVTG3RPFVXWLMWNHB/action/citation_signature","submit_replication":"https://pith.science/pith/RYKSTGVQBCVTG3RPFVXWLMWNHB/action/replication_record"}},"created_at":"2026-05-18T01:06:22.983761+00:00","updated_at":"2026-05-18T01:06:22.983761+00:00"}