{"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"}