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Let $\\Psi$ be a possibly nonmeasurable function from $\\Omega_1$ to $\\R$, and let $X_n(\\omega) = \\Psi(\\Xi_n(\\omega))$. Then we can think of ${X_n}$ as a sequence of independent but possibly nonmeasurable random variables on $\\Omega$. Let $S_n = X_1+...+X_n$. By the ordinary Strong Law of Large Numbers, we almost surely have $E_*[X_1] \\le \\liminf S_n/n \\le \\limsup S_n/n \\le E^*[X_1]$, whe"},"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":"1208.3187","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-08-15T19:37:53Z","cross_cats_sorted":[],"title_canon_sha256":"35020ce8b14d6c5c8e98b61a7884f6945b16fa639d313f83a83f0e1773095ebd","abstract_canon_sha256":"669d76751b451ce56248a82662262bcd08ff8e2382afe76d028a83657285bddd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:47.936797Z","signature_b64":"HGhLqe1g4kCkCQuP3kevw8fQN1i6IGLGZdxwfh5mjW+BrmvbXWnxStHSEZgQqGrELm0kqh/ylhMy9Ue3khQ9CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0652abd70a0708a4caf488404924db646c72e8328a933db6a16f2ea7918f2ddd","last_reissued_at":"2026-05-18T02:44:47.936388Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:47.936388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Law of Large Numbers for Nonmeasurable Identically Distributed Random Variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander R. Pruss","submitted_at":"2012-08-15T19:37:53Z","abstract_excerpt":"Let $\\Omega$ be a countable infinite product $\\Omega^\\N$ of copies of the same probability space $\\Omega_1$, and let ${\\Xi_n}$ be the sequence of the coordinate projection functions from $\\Omega$ to $\\Omega_1$. Let $\\Psi$ be a possibly nonmeasurable function from $\\Omega_1$ to $\\R$, and let $X_n(\\omega) = \\Psi(\\Xi_n(\\omega))$. Then we can think of ${X_n}$ as a sequence of independent but possibly nonmeasurable random variables on $\\Omega$. Let $S_n = X_1+...+X_n$. By the ordinary Strong Law of Large Numbers, we almost surely have $E_*[X_1] \\le \\liminf S_n/n \\le \\limsup S_n/n \\le E^*[X_1]$, whe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.3187","kind":"arxiv","version":2},"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":"1208.3187","created_at":"2026-05-18T02:44:47.936447+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.3187v2","created_at":"2026-05-18T02:44:47.936447+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.3187","created_at":"2026-05-18T02:44:47.936447+00:00"},{"alias_kind":"pith_short_12","alias_value":"AZJKXVYKA4EK","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_16","alias_value":"AZJKXVYKA4EKJSXU","created_at":"2026-05-18T12:26:58.693483+00:00"},{"alias_kind":"pith_short_8","alias_value":"AZJKXVYK","created_at":"2026-05-18T12:26:58.693483+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/AZJKXVYKA4EKJSXURBAESJG3MR","json":"https://pith.science/pith/AZJKXVYKA4EKJSXURBAESJG3MR.json","graph_json":"https://pith.science/api/pith-number/AZJKXVYKA4EKJSXURBAESJG3MR/graph.json","events_json":"https://pith.science/api/pith-number/AZJKXVYKA4EKJSXURBAESJG3MR/events.json","paper":"https://pith.science/paper/AZJKXVYK"},"agent_actions":{"view_html":"https://pith.science/pith/AZJKXVYKA4EKJSXURBAESJG3MR","download_json":"https://pith.science/pith/AZJKXVYKA4EKJSXURBAESJG3MR.json","view_paper":"https://pith.science/paper/AZJKXVYK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.3187&json=true","fetch_graph":"https://pith.science/api/pith-number/AZJKXVYKA4EKJSXURBAESJG3MR/graph.json","fetch_events":"https://pith.science/api/pith-number/AZJKXVYKA4EKJSXURBAESJG3MR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AZJKXVYKA4EKJSXURBAESJG3MR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AZJKXVYKA4EKJSXURBAESJG3MR/action/storage_attestation","attest_author":"https://pith.science/pith/AZJKXVYKA4EKJSXURBAESJG3MR/action/author_attestation","sign_citation":"https://pith.science/pith/AZJKXVYKA4EKJSXURBAESJG3MR/action/citation_signature","submit_replication":"https://pith.science/pith/AZJKXVYKA4EKJSXURBAESJG3MR/action/replication_record"}},"created_at":"2026-05-18T02:44:47.936447+00:00","updated_at":"2026-05-18T02:44:47.936447+00:00"}