{"paper":{"title":"Measurable events indexed by trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Konstantinos Tyros, Pandelis Dodos, Vassilis Kanellopoulos","submitted_at":"2011-05-12T10:42:52Z","abstract_excerpt":"A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\\geq 2$, called the branching number of $T$, such that every $t\\in T$ has exactly $b$ immediate successors. We study the behavior of measurable events in probability spaces indexed by homogeneous trees.\n  Precisely, we show that for every integer $b\\geq 2$ and every integer $n\\geq 1$ there exists an integer $q(b,n)$ with the following property. If $T$ is a homogeneous tree with branching number $b$ and $\\{A_t:t\\in T\\}$ is a family of measurable events in a probability space $(\\Omega,\\Sigma,\\mu)$ satisfy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.2417","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"}