{"paper":{"title":"A covering theorem and the random-indestructibility of the density zero ideal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.LO","authors_text":"M\\'arton Elekes","submitted_at":"2011-09-22T15:52:02Z","abstract_excerpt":"The main goal of this note is to prove the following theorem. If $A_n$ is a sequence of measurable sets in a $\\sigma$-finite measure space $(X, \\mathcal{A}, \\mu)$ that covers $\\mu$-a.e. $x \\in X$ infinitely many times, then there exists a sequence of integers $n_i$ of density zero so that $A_{n_i}$ still covers $\\mu$-a.e. $x \\in X$ infinitely many times. The proof is a probabilistic construction.\n  As an application we give a simple direct proof of the known theorem that the ideal of density zero subsets of the natural numbers is random-indestructible, that is, random forcing does not add a co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4858","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"}