{"paper":{"title":"An ergodic theorem for non-invariant measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Fernando Moreira, Maria Carvalho","submitted_at":"2012-03-27T15:23:25Z","abstract_excerpt":"Given a space $X$, a $\\sigma$-algebra $\\mathfrak{B}$ on $X$ and a measurable map $T:X \\to X$, we say that a measure $\\mu$ is half-invariant if, for any $B \\in \\mathfrak{B}$, we have $\\mu(T^{-1}(B)\\leq \\mu (B)$. In this note we present a generalization of Birkhoff's Ergodic theorem to $\\sigma$-finite half-invariant measures."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.6000","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"}